Properties

Label 32-637e16-1.1-c1e16-0-3
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·4-s − 16·9-s + 15·16-s − 36·23-s + 18·25-s + 36·29-s − 96·36-s − 36·43-s + 12·53-s + 36·79-s + 120·81-s − 216·92-s + 108·100-s + 48·107-s + 60·113-s + 216·116-s + 84·121-s + 127-s + 131-s + 137-s + 139-s − 240·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 3·4-s − 5.33·9-s + 15/4·16-s − 7.50·23-s + 18/5·25-s + 6.68·29-s − 16·36-s − 5.48·43-s + 1.64·53-s + 4.05·79-s + 40/3·81-s − 22.5·92-s + 54/5·100-s + 4.64·107-s + 5.64·113-s + 20.0·116-s + 7.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 20·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-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

\(L(1)\) \(\approx\) \(21.12245373\)
\(L(\frac12)\) \(\approx\) \(21.12245373\)
\(L(\frac{3}{2})\) 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 \( 1 + 22 T^{2} + 160 T^{4} - 1766 T^{6} - 47906 T^{8} - 1766 p^{2} T^{10} + 160 p^{4} T^{12} + 22 p^{6} T^{14} + p^{8} T^{16} \)
good2 \( ( 1 - 3 T^{2} + 3 p T^{4} - 5 p T^{8} + 3 p^{5} T^{12} - 3 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
3 \( ( 1 + 8 T^{2} + 4 p^{2} T^{4} + 134 T^{6} + 442 T^{8} + 134 p^{2} T^{10} + 4 p^{6} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
5 \( ( 1 - 9 T^{2} + 117 T^{4} - 678 T^{6} + 4592 T^{8} - 678 p^{2} T^{10} + 117 p^{4} T^{12} - 9 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
11 \( ( 1 - 42 T^{2} + 954 T^{4} - 14424 T^{6} + 15874 p T^{8} - 14424 p^{2} T^{10} + 954 p^{4} T^{12} - 42 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
17 \( ( 1 + 58 T^{2} + 2044 T^{4} + 51244 T^{6} + 1002934 T^{8} + 51244 p^{2} T^{10} + 2044 p^{4} T^{12} + 58 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( ( 1 - 77 T^{2} + 3421 T^{4} - 103988 T^{6} + 2284354 T^{8} - 103988 p^{2} T^{10} + 3421 p^{4} T^{12} - 77 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
23 \( ( 1 + 9 T + 53 T^{2} + 180 T^{3} + 588 T^{4} + 180 p T^{5} + 53 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
29 \( ( 1 - 9 T + 95 T^{2} - 18 p T^{3} + 3684 T^{4} - 18 p^{2} T^{5} + 95 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
31 \( ( 1 - 83 T^{2} + 5695 T^{4} - 241382 T^{6} + 8942926 T^{8} - 241382 p^{2} T^{10} + 5695 p^{4} T^{12} - 83 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
37 \( ( 1 - 206 T^{2} + 20746 T^{4} - 1326044 T^{6} + 58452550 T^{8} - 1326044 p^{2} T^{10} + 20746 p^{4} T^{12} - 206 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
41 \( ( 1 - 174 T^{2} + 312 p T^{4} - 13218 p T^{6} + 19824686 T^{8} - 13218 p^{3} T^{10} + 312 p^{5} T^{12} - 174 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( ( 1 + 9 T + 151 T^{2} + 1116 T^{3} + 9360 T^{4} + 1116 p T^{5} + 151 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
47 \( ( 1 - 261 T^{2} + 32325 T^{4} - 2543628 T^{6} + 140726834 T^{8} - 2543628 p^{2} T^{10} + 32325 p^{4} T^{12} - 261 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( ( 1 - 3 T + 107 T^{2} - 270 T^{3} + 8226 T^{4} - 270 p T^{5} + 107 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
59 \( ( 1 - 282 T^{2} + 41940 T^{4} - 4081338 T^{6} + 282790766 T^{8} - 4081338 p^{2} T^{10} + 41940 p^{4} T^{12} - 282 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
61 \( ( 1 + 320 T^{2} + 50548 T^{4} + 5169440 T^{6} + 372288646 T^{8} + 5169440 p^{2} T^{10} + 50548 p^{4} T^{12} + 320 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
67 \( ( 1 - 188 T^{2} + 11284 T^{4} + 133180 T^{6} - 45913034 T^{8} + 133180 p^{2} T^{10} + 11284 p^{4} T^{12} - 188 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
71 \( ( 1 - 444 T^{2} + 92718 T^{4} - 11864022 T^{6} + 1016978846 T^{8} - 11864022 p^{2} T^{10} + 92718 p^{4} T^{12} - 444 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
73 \( ( 1 - 395 T^{2} + 76195 T^{4} - 9360704 T^{6} + 805618420 T^{8} - 9360704 p^{2} T^{10} + 76195 p^{4} T^{12} - 395 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
79 \( ( 1 - 9 T + 327 T^{2} - 2088 T^{3} + 39140 T^{4} - 2088 p T^{5} + 327 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
83 \( ( 1 - 507 T^{2} + 117435 T^{4} - 16687278 T^{6} + 1636969526 T^{8} - 16687278 p^{2} T^{10} + 117435 p^{4} T^{12} - 507 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
89 \( ( 1 - 327 T^{2} + 64371 T^{4} - 8894988 T^{6} + 900996404 T^{8} - 8894988 p^{2} T^{10} + 64371 p^{4} T^{12} - 327 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( ( 1 - 239 T^{2} + 43123 T^{4} - 5980772 T^{6} + 636722164 T^{8} - 5980772 p^{2} T^{10} + 43123 p^{4} T^{12} - 239 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
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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

−2.90870078349666188480153496777, −2.89313628132265685974083367873, −2.64369698217787594032230661333, −2.63552976307937293773089309113, −2.52511065860498504359334667506, −2.48080360854641910650735168198, −2.22987508235861721673327092508, −2.15756853696236106719346247699, −2.09303030791743850628126878799, −2.08765232840333561431705794195, −1.97445200829911135925852425046, −1.96481634824069799610228170202, −1.90224961669680199924580662220, −1.90144432928700038317581042102, −1.71867153774401612596302557481, −1.61762516249177544862296570984, −1.47692629579325618155691201148, −1.34021620224077051654192141902, −1.00361730339868902955540741875, −0.840722086822555099916236828563, −0.69807622465968078829108985231, −0.65443836029599994622976451237, −0.50104839014721444935930035914, −0.44666491403701600205964716818, −0.38189717922567172418118627188, 0.38189717922567172418118627188, 0.44666491403701600205964716818, 0.50104839014721444935930035914, 0.65443836029599994622976451237, 0.69807622465968078829108985231, 0.840722086822555099916236828563, 1.00361730339868902955540741875, 1.34021620224077051654192141902, 1.47692629579325618155691201148, 1.61762516249177544862296570984, 1.71867153774401612596302557481, 1.90144432928700038317581042102, 1.90224961669680199924580662220, 1.96481634824069799610228170202, 1.97445200829911135925852425046, 2.08765232840333561431705794195, 2.09303030791743850628126878799, 2.15756853696236106719346247699, 2.22987508235861721673327092508, 2.48080360854641910650735168198, 2.52511065860498504359334667506, 2.63552976307937293773089309113, 2.64369698217787594032230661333, 2.89313628132265685974083367873, 2.90870078349666188480153496777

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

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