Properties

Label 32-637e16-1.1-c1e16-0-2
Degree $32$
Conductor $7.349\times 10^{44}$
Sign $1$
Analytic cond. $2.00754\times 10^{11}$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 10·4-s + 32·8-s − 20·9-s + 8·11-s + 87·16-s − 80·18-s + 32·22-s + 12·23-s + 26·25-s + 8·29-s + 196·32-s − 200·36-s − 8·37-s + 32·43-s + 80·44-s + 48·46-s + 104·50-s + 4·53-s + 32·58-s + 438·64-s − 40·67-s + 8·71-s − 640·72-s − 32·74-s + 4·79-s + 166·81-s + ⋯
L(s)  = 1  + 2.82·2-s + 5·4-s + 11.3·8-s − 6.66·9-s + 2.41·11-s + 87/4·16-s − 18.8·18-s + 6.82·22-s + 2.50·23-s + 26/5·25-s + 1.48·29-s + 34.6·32-s − 33.3·36-s − 1.31·37-s + 4.87·43-s + 12.0·44-s + 7.07·46-s + 14.7·50-s + 0.549·53-s + 4.20·58-s + 54.7·64-s − 4.88·67-s + 0.949·71-s − 75.4·72-s − 3.71·74-s + 0.450·79-s + 18.4·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(7^{32} \cdot 13^{16}\)
Sign: $1$
Analytic conductor: \(2.00754\times 10^{11}\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 7^{32} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(226.8379804\)
\(L(\frac12)\) \(\approx\) \(226.8379804\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + 34 T^{2} + 430 T^{4} + 4984 T^{6} + 76567 T^{8} + 4984 p^{2} T^{10} + 430 p^{4} T^{12} + 34 p^{6} T^{14} + p^{8} T^{16} \)
good2 \( ( 1 - p T + T^{2} - 3 p T^{3} + 5 p T^{4} - p^{2} T^{5} + 27 T^{6} - 5 p^{3} T^{7} + 13 T^{8} - 5 p^{4} T^{9} + 27 p^{2} T^{10} - p^{5} T^{11} + 5 p^{5} T^{12} - 3 p^{6} T^{13} + p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} )^{2} \)
3 \( ( 1 + 10 T^{2} + 67 T^{4} + 34 p^{2} T^{6} + 1057 T^{8} + 34 p^{4} T^{10} + 67 p^{4} T^{12} + 10 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
5 \( 1 - 26 T^{2} + 73 p T^{4} - 674 p T^{6} + 22132 T^{8} - 101676 T^{10} + 285823 T^{12} - 109698 T^{14} - 2335389 T^{16} - 109698 p^{2} T^{18} + 285823 p^{4} T^{20} - 101676 p^{6} T^{22} + 22132 p^{8} T^{24} - 674 p^{11} T^{26} + 73 p^{13} T^{28} - 26 p^{14} T^{30} + p^{16} T^{32} \)
11 \( ( 1 - 2 T + 39 T^{2} - 54 T^{3} + 611 T^{4} - 54 p T^{5} + 39 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
17 \( 1 - 78 T^{2} + 2942 T^{4} - 74036 T^{6} + 1506893 T^{8} - 29044204 T^{10} + 585942562 T^{12} - 11972588178 T^{14} + 220591800628 T^{16} - 11972588178 p^{2} T^{18} + 585942562 p^{4} T^{20} - 29044204 p^{6} T^{22} + 1506893 p^{8} T^{24} - 74036 p^{10} T^{26} + 2942 p^{12} T^{28} - 78 p^{14} T^{30} + p^{16} T^{32} \)
19 \( ( 1 + 58 T^{2} + 1703 T^{4} + 44478 T^{6} + 996825 T^{8} + 44478 p^{2} T^{10} + 1703 p^{4} T^{12} + 58 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
23 \( ( 1 - 6 T - 42 T^{2} + 248 T^{3} + 1306 T^{4} - 4598 T^{5} - 43856 T^{6} + 22390 T^{7} + 1346875 T^{8} + 22390 p T^{9} - 43856 p^{2} T^{10} - 4598 p^{3} T^{11} + 1306 p^{4} T^{12} + 248 p^{5} T^{13} - 42 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
29 \( ( 1 - 4 T - 15 T^{2} + 192 T^{3} - 1568 T^{4} + 5696 T^{5} - p T^{6} - 217482 T^{7} + 2139619 T^{8} - 217482 p T^{9} - p^{3} T^{10} + 5696 p^{3} T^{11} - 1568 p^{4} T^{12} + 192 p^{5} T^{13} - 15 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
31 \( 1 - 178 T^{2} + 16278 T^{4} - 1070764 T^{6} + 57589565 T^{8} - 2645809244 T^{10} + 106521129450 T^{12} - 3844760620262 T^{14} + 125420674945492 T^{16} - 3844760620262 p^{2} T^{18} + 106521129450 p^{4} T^{20} - 2645809244 p^{6} T^{22} + 57589565 p^{8} T^{24} - 1070764 p^{10} T^{26} + 16278 p^{12} T^{28} - 178 p^{14} T^{30} + p^{16} T^{32} \)
37 \( ( 1 + 4 T - 46 T^{2} + 152 T^{3} + 2182 T^{4} - 7008 T^{5} + 73816 T^{6} + 392460 T^{7} - 3440049 T^{8} + 392460 p T^{9} + 73816 p^{2} T^{10} - 7008 p^{3} T^{11} + 2182 p^{4} T^{12} + 152 p^{5} T^{13} - 46 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
41 \( 1 - 152 T^{2} + 16324 T^{4} - 1214800 T^{6} + 75014282 T^{8} - 3793839848 T^{10} + 172337071760 T^{12} - 7119768845032 T^{14} + 293658219524371 T^{16} - 7119768845032 p^{2} T^{18} + 172337071760 p^{4} T^{20} - 3793839848 p^{6} T^{22} + 75014282 p^{8} T^{24} - 1214800 p^{10} T^{26} + 16324 p^{12} T^{28} - 152 p^{14} T^{30} + p^{16} T^{32} \)
43 \( ( 1 - 16 T + 99 T^{2} - 472 T^{3} + 1730 T^{4} + 7168 T^{5} - 116817 T^{6} + 1124260 T^{7} - 9477617 T^{8} + 1124260 p T^{9} - 116817 p^{2} T^{10} + 7168 p^{3} T^{11} + 1730 p^{4} T^{12} - 472 p^{5} T^{13} + 99 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
47 \( 1 - 150 T^{2} + 11006 T^{4} - 308228 T^{6} - 11259283 T^{8} + 1453428692 T^{10} - 48584589662 T^{12} - 402313000122 T^{14} + 91917552255220 T^{16} - 402313000122 p^{2} T^{18} - 48584589662 p^{4} T^{20} + 1453428692 p^{6} T^{22} - 11259283 p^{8} T^{24} - 308228 p^{10} T^{26} + 11006 p^{12} T^{28} - 150 p^{14} T^{30} + p^{16} T^{32} \)
53 \( ( 1 - 2 T - 128 T^{2} + 232 T^{3} + 8283 T^{4} - 11300 T^{5} - 364348 T^{6} + 286726 T^{7} + 15832832 T^{8} + 286726 p T^{9} - 364348 p^{2} T^{10} - 11300 p^{3} T^{11} + 8283 p^{4} T^{12} + 232 p^{5} T^{13} - 128 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
59 \( 1 - 278 T^{2} + 40998 T^{4} - 3814004 T^{6} + 237186965 T^{8} - 9386021884 T^{10} + 154854604410 T^{12} + 7674718497878 T^{14} - 788274714787388 T^{16} + 7674718497878 p^{2} T^{18} + 154854604410 p^{4} T^{20} - 9386021884 p^{6} T^{22} + 237186965 p^{8} T^{24} - 3814004 p^{10} T^{26} + 40998 p^{12} T^{28} - 278 p^{14} T^{30} + p^{16} T^{32} \)
61 \( ( 1 + 380 T^{2} + 67856 T^{4} + 7443748 T^{6} + 547986382 T^{8} + 7443748 p^{2} T^{10} + 67856 p^{4} T^{12} + 380 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
67 \( ( 1 + 10 T + 262 T^{2} + 1788 T^{3} + 25847 T^{4} + 1788 p T^{5} + 262 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
71 \( ( 1 - 4 T - 102 T^{2} + 696 T^{3} - 1042 T^{4} - 9336 T^{5} - 125464 T^{6} - 1401412 T^{7} + 51152511 T^{8} - 1401412 p T^{9} - 125464 p^{2} T^{10} - 9336 p^{3} T^{11} - 1042 p^{4} T^{12} + 696 p^{5} T^{13} - 102 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
73 \( 1 - 156 T^{2} + 17104 T^{4} - 833272 T^{6} + 18044674 T^{8} + 1658812620 T^{10} + 228525055904 T^{12} - 629317998452 p T^{14} + 5084762171268243 T^{16} - 629317998452 p^{3} T^{18} + 228525055904 p^{4} T^{20} + 1658812620 p^{6} T^{22} + 18044674 p^{8} T^{24} - 833272 p^{10} T^{26} + 17104 p^{12} T^{28} - 156 p^{14} T^{30} + p^{16} T^{32} \)
79 \( ( 1 - 2 T - 34 T^{2} - 552 T^{3} + 234 T^{4} + 34590 T^{5} + 504496 T^{6} - 2488078 T^{7} - 45650053 T^{8} - 2488078 p T^{9} + 504496 p^{2} T^{10} + 34590 p^{3} T^{11} + 234 p^{4} T^{12} - 552 p^{5} T^{13} - 34 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
83 \( ( 1 + 314 T^{2} + 46430 T^{4} + 4837464 T^{6} + 427877847 T^{8} + 4837464 p^{2} T^{10} + 46430 p^{4} T^{12} + 314 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
89 \( 1 - 106 T^{2} - 6635 T^{4} + 2260150 T^{6} - 56645140 T^{8} - 20540045260 T^{10} + 2028500040815 T^{12} + 92860376641870 T^{14} - 22701882367538285 T^{16} + 92860376641870 p^{2} T^{18} + 2028500040815 p^{4} T^{20} - 20540045260 p^{6} T^{22} - 56645140 p^{8} T^{24} + 2260150 p^{10} T^{26} - 6635 p^{12} T^{28} - 106 p^{14} T^{30} + p^{16} T^{32} \)
97 \( 1 - 412 T^{2} + 87509 T^{4} - 11349760 T^{6} + 907888036 T^{8} - 32651756224 T^{10} - 1974791695085 T^{12} + 422191622326362 T^{14} - 45739383615060693 T^{16} + 422191622326362 p^{2} T^{18} - 1974791695085 p^{4} T^{20} - 32651756224 p^{6} T^{22} + 907888036 p^{8} T^{24} - 11349760 p^{10} T^{26} + 87509 p^{12} T^{28} - 412 p^{14} T^{30} + p^{16} T^{32} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.73676171278023224283348467809, −2.70201697783377317944785532007, −2.67951226208558297893152438853, −2.62962589377810023951117722675, −2.61810552408911526246420811017, −2.54604574472280387541891876815, −2.54397241791588699981131003131, −2.48272078879581929991038210113, −2.43700054459794488032970672384, −2.31397630469855369914719347403, −1.96102416199171339369350139286, −1.80085582648900074099345073481, −1.78371162756650137901126562898, −1.53643566161148542785053123402, −1.51546544820126008175384224121, −1.47003209809559148518823047527, −1.36717629302354125058642966541, −1.35976748940641493091030517822, −1.27037103182144382545279948801, −1.07127041897972966231277231557, −0.999022790644618120248558795987, −0.893025725353493723907307387666, −0.61498371685413355659004989304, −0.43554863557981812431947576478, −0.36453354893281442051376943767, 0.36453354893281442051376943767, 0.43554863557981812431947576478, 0.61498371685413355659004989304, 0.893025725353493723907307387666, 0.999022790644618120248558795987, 1.07127041897972966231277231557, 1.27037103182144382545279948801, 1.35976748940641493091030517822, 1.36717629302354125058642966541, 1.47003209809559148518823047527, 1.51546544820126008175384224121, 1.53643566161148542785053123402, 1.78371162756650137901126562898, 1.80085582648900074099345073481, 1.96102416199171339369350139286, 2.31397630469855369914719347403, 2.43700054459794488032970672384, 2.48272078879581929991038210113, 2.54397241791588699981131003131, 2.54604574472280387541891876815, 2.61810552408911526246420811017, 2.62962589377810023951117722675, 2.67951226208558297893152438853, 2.70201697783377317944785532007, 2.73676171278023224283348467809

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.