Properties

Label 32-637e16-1.1-c1e16-0-0
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  − 8·2-s + 28·4-s − 40·8-s + 10·9-s − 4·11-s − 66·16-s − 80·18-s + 32·22-s − 24·23-s + 26·25-s + 8·29-s + 448·32-s + 280·36-s + 16·37-s + 32·43-s − 112·44-s + 192·46-s − 208·50-s + 4·53-s − 64·58-s − 912·64-s + 20·67-s + 8·71-s − 400·72-s − 128·74-s + 4·79-s + 67·81-s + ⋯
L(s)  = 1  − 5.65·2-s + 14·4-s − 14.1·8-s + 10/3·9-s − 1.20·11-s − 16.5·16-s − 18.8·18-s + 6.82·22-s − 5.00·23-s + 26/5·25-s + 1.48·29-s + 79.1·32-s + 46.6·36-s + 2.63·37-s + 4.87·43-s − 16.8·44-s + 28.3·46-s − 29.4·50-s + 0.549·53-s − 8.40·58-s − 114·64-s + 2.44·67-s + 0.949·71-s − 47.1·72-s − 14.8·74-s + 0.450·79-s + 67/9·81-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: induced by $\chi_{637} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 7^{32} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2893341587\)
\(L(\frac12)\) \(\approx\) \(0.2893341587\)
\(L(\frac{3}{2})\) 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 \( ( 1 + p T + 3 T^{2} - T^{4} + 3 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} )^{4} \)
3 \( 1 - 10 T^{2} + 11 p T^{4} - 58 T^{6} + 124 p T^{8} - 2116 T^{10} + 6131 T^{12} - 1774 p^{2} T^{14} + 50155 T^{16} - 1774 p^{4} T^{18} + 6131 p^{4} T^{20} - 2116 p^{6} T^{22} + 124 p^{9} T^{24} - 58 p^{10} T^{26} + 11 p^{13} T^{28} - 10 p^{14} T^{30} + p^{16} T^{32} \)
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 \( ( 1 + 2 T - 35 T^{2} - 30 T^{3} + 802 T^{4} + 256 T^{5} - 12663 T^{6} - 1238 T^{7} + 153211 T^{8} - 1238 p T^{9} - 12663 p^{2} T^{10} + 256 p^{3} T^{11} + 802 p^{4} T^{12} - 30 p^{5} T^{13} - 35 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
17 \( ( 1 + 78 T^{2} + 3142 T^{4} + 85520 T^{6} + 1694711 T^{8} + 85520 p^{2} T^{10} + 3142 p^{4} T^{12} + 78 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( 1 - 58 T^{2} + 1661 T^{4} - 9818 T^{6} - 676340 T^{8} + 23829108 T^{10} - 206707529 T^{12} - 5249133354 T^{14} + 201079481139 T^{16} - 5249133354 p^{2} T^{18} - 206707529 p^{4} T^{20} + 23829108 p^{6} T^{22} - 676340 p^{8} T^{24} - 9818 p^{10} T^{26} + 1661 p^{12} T^{28} - 58 p^{14} T^{30} + p^{16} T^{32} \)
23 \( ( 1 + 6 T + 78 T^{2} + 358 T^{3} + 2630 T^{4} + 358 p T^{5} + 78 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
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 \( ( 1 - 4 T + 62 T^{2} - 48 T^{3} + 1470 T^{4} - 48 p T^{5} + 62 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
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 \( ( 1 + 278 T^{2} + 36286 T^{4} + 3136752 T^{6} + 207469775 T^{8} + 3136752 p^{2} T^{10} + 36286 p^{4} T^{12} + 278 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
61 \( 1 - 380 T^{2} + 76544 T^{4} - 10897784 T^{6} + 1227826114 T^{8} - 116331500276 T^{10} + 9578956327264 T^{12} - 696692904086100 T^{14} + 45045833784892083 T^{16} - 696692904086100 p^{2} T^{18} + 9578956327264 p^{4} T^{20} - 116331500276 p^{6} T^{22} + 1227826114 p^{8} T^{24} - 10897784 p^{10} T^{26} + 76544 p^{12} T^{28} - 380 p^{14} T^{30} + p^{16} T^{32} \)
67 \( ( 1 - 10 T - 162 T^{2} + 956 T^{3} + 24917 T^{4} - 71312 T^{5} - 2420694 T^{6} + 1789858 T^{7} + 185730724 T^{8} + 1789858 p T^{9} - 2420694 p^{2} T^{10} - 71312 p^{3} T^{11} + 24917 p^{4} T^{12} + 956 p^{5} T^{13} - 162 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
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 \( ( 1 + 106 T^{2} + 17871 T^{4} + 2077238 T^{6} + 155830553 T^{8} + 2077238 p^{2} T^{10} + 17871 p^{4} T^{12} + 106 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
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

−2.67339105952063251936383461332, −2.65695754868378917317304619241, −2.51896304827374175188699923491, −2.40748614510542649189598514245, −2.14701931989856412948559058917, −2.14002020018408487611894547576, −2.12906343988599297098009574300, −2.10360427796975208794920637830, −2.01681506639517474155572396891, −1.95099558375382084106402016846, −1.93736358616088475201393040636, −1.87586627491988705002830840142, −1.72888373508026676429052858210, −1.60351469940836083128015690356, −1.32204334879384365880013536142, −1.23945567615879988125092159685, −1.11492703540708945793058665478, −1.09219037361075807029991043589, −1.05844891543048948426270760138, −0.878688011360595874755261321742, −0.76423655163299994550154811930, −0.74268881960285116168918604274, −0.72273975159116333587483893882, −0.35894710697158135918754691277, −0.15970903169895791234650601035, 0.15970903169895791234650601035, 0.35894710697158135918754691277, 0.72273975159116333587483893882, 0.74268881960285116168918604274, 0.76423655163299994550154811930, 0.878688011360595874755261321742, 1.05844891543048948426270760138, 1.09219037361075807029991043589, 1.11492703540708945793058665478, 1.23945567615879988125092159685, 1.32204334879384365880013536142, 1.60351469940836083128015690356, 1.72888373508026676429052858210, 1.87586627491988705002830840142, 1.93736358616088475201393040636, 1.95099558375382084106402016846, 2.01681506639517474155572396891, 2.10360427796975208794920637830, 2.12906343988599297098009574300, 2.14002020018408487611894547576, 2.14701931989856412948559058917, 2.40748614510542649189598514245, 2.51896304827374175188699923491, 2.65695754868378917317304619241, 2.67339105952063251936383461332

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

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