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Properties

Label 32-2340e16-1.1-c1e16-0-1
Degree $32$
Conductor $8.081\times 10^{53}$
Sign $1$
Analytic cond. $2.20743\times 10^{20}$
Root an. cond. $4.32261$
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  + 16·7-s + 8·13-s + 12·25-s − 32·31-s − 16·37-s + 16·43-s + 48·49-s − 16·61-s + 128·91-s − 16·103-s − 32·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 16·169-s + 173-s + 192·175-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 6.04·7-s + 2.21·13-s + 12/5·25-s − 5.74·31-s − 2.63·37-s + 2.43·43-s + 48/7·49-s − 2.04·61-s + 13.4·91-s − 1.57·103-s − 3.06·109-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.23·169-s + 0.0760·173-s + 14.5·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{32} \cdot 5^{16} \cdot 13^{16}\)
Sign: $1$
Analytic conductor: \(2.20743\times 10^{20}\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{32} \cdot 5^{16} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(21.58015241\)
\(L(\frac12)\) \(\approx\) \(21.58015241\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - 12 T^{2} + 88 T^{4} - 108 p T^{6} + 114 p^{2} T^{8} - 108 p^{3} T^{10} + 88 p^{4} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} \)
13 \( ( 1 - 4 T + 16 T^{2} - 4 T^{3} + 82 T^{4} - 4 p T^{5} + 16 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
good7 \( ( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{8} \)
11 \( 1 - 256 T^{4} + 40420 T^{8} - 4184320 T^{12} + 4087606 p^{2} T^{16} - 4184320 p^{4} T^{20} + 40420 p^{8} T^{24} - 256 p^{12} T^{28} + p^{16} T^{32} \)
17 \( 1 + 968 T^{4} + 633244 T^{8} + 271156856 T^{12} + 91485322822 T^{16} + 271156856 p^{4} T^{20} + 633244 p^{8} T^{24} + 968 p^{12} T^{28} + p^{16} T^{32} \)
19 \( ( 1 - 48 T^{3} + 220 T^{4} + 1584 T^{5} + 1152 T^{6} + 12192 T^{7} - 149082 T^{8} + 12192 p T^{9} + 1152 p^{2} T^{10} + 1584 p^{3} T^{11} + 220 p^{4} T^{12} - 48 p^{5} T^{13} + p^{8} T^{16} )^{2} \)
23 \( 1 + 392 T^{4} + 185116 T^{8} + 141338552 T^{12} + 61961566918 T^{16} + 141338552 p^{4} T^{20} + 185116 p^{8} T^{24} + 392 p^{12} T^{28} + p^{16} T^{32} \)
29 \( ( 1 - 72 T^{2} + 4156 T^{4} - 167352 T^{6} + 5587878 T^{8} - 167352 p^{2} T^{10} + 4156 p^{4} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
31 \( ( 1 + 16 T + 128 T^{2} + 896 T^{3} + 4988 T^{4} + 18080 T^{5} + 52224 T^{6} + 38832 T^{7} - 727994 T^{8} + 38832 p T^{9} + 52224 p^{2} T^{10} + 18080 p^{3} T^{11} + 4988 p^{4} T^{12} + 896 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
37 \( ( 1 + 4 T + 88 T^{2} + 244 T^{3} + 4066 T^{4} + 244 p T^{5} + 88 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
41 \( 1 - 2848 T^{4} + 7074724 T^{8} - 11554866016 T^{12} + 17670684392902 T^{16} - 11554866016 p^{4} T^{20} + 7074724 p^{8} T^{24} - 2848 p^{12} T^{28} + p^{16} T^{32} \)
43 \( ( 1 - 8 T + 32 T^{2} - 136 T^{3} - 772 T^{4} + 824 T^{5} + 27360 T^{6} - 379272 T^{7} + 4778854 T^{8} - 379272 p T^{9} + 27360 p^{2} T^{10} + 824 p^{3} T^{11} - 772 p^{4} T^{12} - 136 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
47 \( ( 1 + 156 T^{2} + 15880 T^{4} + 1133772 T^{6} + 60514242 T^{8} + 1133772 p^{2} T^{10} + 15880 p^{4} T^{12} + 156 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( 1 + 3656 T^{4} + 29518300 T^{8} + 80188267640 T^{12} + 343415335738246 T^{16} + 80188267640 p^{4} T^{20} + 29518300 p^{8} T^{24} + 3656 p^{12} T^{28} + p^{16} T^{32} \)
59 \( 1 - 9856 T^{4} + 61399780 T^{8} - 251593916800 T^{12} + 942726012976006 T^{16} - 251593916800 p^{4} T^{20} + 61399780 p^{8} T^{24} - 9856 p^{12} T^{28} + p^{16} T^{32} \)
61 \( ( 1 + 4 T + 88 T^{2} + 340 T^{3} + 3778 T^{4} + 340 p T^{5} + 88 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
67 \( ( 1 - 104 T^{2} + 13348 T^{4} - 1259960 T^{6} + 78672934 T^{8} - 1259960 p^{2} T^{10} + 13348 p^{4} T^{12} - 104 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
71 \( 1 + 4928 T^{4} + 51072676 T^{8} + 212553142208 T^{12} + 1886637510937798 T^{16} + 212553142208 p^{4} T^{20} + 51072676 p^{8} T^{24} + 4928 p^{12} T^{28} + p^{16} T^{32} \)
73 \( ( 1 - 32 T^{2} + 7012 T^{4} - 278624 T^{6} + 32055814 T^{8} - 278624 p^{2} T^{10} + 7012 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
79 \( ( 1 - 448 T^{2} + 94468 T^{4} - 12558400 T^{6} + 1168950406 T^{8} - 12558400 p^{2} T^{10} + 94468 p^{4} T^{12} - 448 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
83 \( ( 1 + 412 T^{2} + 85864 T^{4} + 11731948 T^{6} + 1141361122 T^{8} + 11731948 p^{2} T^{10} + 85864 p^{4} T^{12} + 412 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
89 \( 1 + 14816 T^{4} + 212856484 T^{8} + 2275545764768 T^{12} + 19566480195765958 T^{16} + 2275545764768 p^{4} T^{20} + 212856484 p^{8} T^{24} + 14816 p^{12} T^{28} + p^{16} T^{32} \)
97 \( ( 1 - 544 T^{2} + 143620 T^{4} - 24043360 T^{6} + 2778027526 T^{8} - 24043360 p^{2} T^{10} + 143620 p^{4} T^{12} - 544 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
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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

−2.04350368900187131029597618738, −2.01998418333969777709342801289, −1.95269531464870001675590256392, −1.88346950273523187955059190202, −1.83166434267692979941244230532, −1.75435354589969771955591631583, −1.72523394596046411940794886802, −1.63242723035388946664335337433, −1.59539284199696911614777148705, −1.57875138659276502395036680638, −1.49923682054533961865374768480, −1.49878601065096539717149049086, −1.43227983029532654322890677686, −1.40042407233861489581749251056, −1.39411274606818221438089680887, −1.16175438205092229217795144088, −1.04136729620434123125440077569, −0.914914380867522252997620657446, −0.77569169966319916603778208741, −0.68537411028253893791275857074, −0.65122131547238315739877127817, −0.55397071773260444878413025628, −0.23712036544655807946930874701, −0.18365167822689752527155078431, −0.17066075678660330995872699503, 0.17066075678660330995872699503, 0.18365167822689752527155078431, 0.23712036544655807946930874701, 0.55397071773260444878413025628, 0.65122131547238315739877127817, 0.68537411028253893791275857074, 0.77569169966319916603778208741, 0.914914380867522252997620657446, 1.04136729620434123125440077569, 1.16175438205092229217795144088, 1.39411274606818221438089680887, 1.40042407233861489581749251056, 1.43227983029532654322890677686, 1.49878601065096539717149049086, 1.49923682054533961865374768480, 1.57875138659276502395036680638, 1.59539284199696911614777148705, 1.63242723035388946664335337433, 1.72523394596046411940794886802, 1.75435354589969771955591631583, 1.83166434267692979941244230532, 1.88346950273523187955059190202, 1.95269531464870001675590256392, 2.01998418333969777709342801289, 2.04350368900187131029597618738

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

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