Properties

Label 4-2738e2-1.1-c1e2-0-1
Degree $4$
Conductor $7496644$
Sign $1$
Analytic cond. $477.992$
Root an. cond. $4.67579$
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  − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 4·7-s − 4·8-s + 6·11-s − 6·12-s + 8·14-s + 5·16-s + 12·17-s − 6·19-s + 8·21-s − 12·22-s + 6·23-s + 8·24-s − 7·25-s + 2·27-s − 12·28-s + 6·31-s − 6·32-s − 12·33-s − 24·34-s + 12·38-s − 6·41-s − 16·42-s + 12·43-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 1.51·7-s − 1.41·8-s + 1.80·11-s − 1.73·12-s + 2.13·14-s + 5/4·16-s + 2.91·17-s − 1.37·19-s + 1.74·21-s − 2.55·22-s + 1.25·23-s + 1.63·24-s − 7/5·25-s + 0.384·27-s − 2.26·28-s + 1.07·31-s − 1.06·32-s − 2.08·33-s − 4.11·34-s + 1.94·38-s − 0.937·41-s − 2.46·42-s + 1.82·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7496644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7496644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7496644\)    =    \(2^{2} \cdot 37^{4}\)
Sign: $1$
Analytic conductor: \(477.992\)
Root analytic conductor: \(4.67579\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7496644,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8235519087\)
\(L(\frac12)\) \(\approx\) \(0.8235519087\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( ( 1 + T )^{2} \)
37 \( 1 \)
good3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_e
5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_bc
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$D_{4}$ \( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.17.am_cp
19$D_{4}$ \( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_bs
23$D_{4}$ \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_ca
29$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.29.a_ar
31$D_{4}$ \( 1 - 6 T + 68 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_cq
41$D_{4}$ \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_br
43$D_{4}$ \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.43.am_cw
47$D_{4}$ \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_dw
53$D_{4}$ \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.53.am_fa
59$D_{4}$ \( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_fm
61$C_2^2$ \( 1 + 119 T^{2} + p^{2} T^{4} \) 2.61.a_ep
67$D_{4}$ \( 1 - 10 T + 132 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.67.ak_fc
71$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \) 2.71.a_fa
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.73.i_gg
79$D_{4}$ \( 1 - 30 T + 380 T^{2} - 30 p T^{3} + p^{2} T^{4} \) 2.79.abe_oq
83$D_{4}$ \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_dw
89$D_{4}$ \( 1 - 24 T + 295 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.89.ay_lj
97$D_{4}$ \( 1 - 12 T + 227 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.97.am_it
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.998277313238057602529727843297, −8.926624225415593607618926150811, −8.162070991483014645161041440095, −7.997665127226760047877062422275, −7.47335134194929935814899708968, −7.13552087114778179693172606096, −6.56723503070155757700697005490, −6.44657680016031144497209973518, −6.05167483312816899559524704054, −5.92145859524902222151916670887, −5.19851201789240020320149469816, −5.07808978182210144486938026560, −3.87665090656973317113783054985, −3.83476848129932883066520274252, −3.43947277139845742920653894740, −2.75060439615281432430041041540, −2.28254180652175867425864850063, −1.47307355369391441349623430751, −0.78061597215469401137474252616, −0.63068951697585718813662734457, 0.63068951697585718813662734457, 0.78061597215469401137474252616, 1.47307355369391441349623430751, 2.28254180652175867425864850063, 2.75060439615281432430041041540, 3.43947277139845742920653894740, 3.83476848129932883066520274252, 3.87665090656973317113783054985, 5.07808978182210144486938026560, 5.19851201789240020320149469816, 5.92145859524902222151916670887, 6.05167483312816899559524704054, 6.44657680016031144497209973518, 6.56723503070155757700697005490, 7.13552087114778179693172606096, 7.47335134194929935814899708968, 7.997665127226760047877062422275, 8.162070991483014645161041440095, 8.926624225415593607618926150811, 8.998277313238057602529727843297

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