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Properties

Label 4-792e2-1.1-c1e2-0-114
Degree $4$
Conductor $627264$
Sign $-1$
Analytic cond. $39.9948$
Root an. cond. $2.51478$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 2·4-s − 2·6-s + 7-s + 9-s − 3·11-s − 2·12-s + 2·14-s − 4·16-s + 2·18-s − 21-s − 6·22-s + 5·25-s − 27-s + 2·28-s − 4·29-s − 8·32-s + 3·33-s + 2·36-s − 2·42-s − 6·44-s + 4·48-s − 13·49-s + 10·50-s − 2·54-s − 8·58-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.577·12-s + 0.534·14-s − 16-s + 0.471·18-s − 0.218·21-s − 1.27·22-s + 25-s − 0.192·27-s + 0.377·28-s − 0.742·29-s − 1.41·32-s + 0.522·33-s + 1/3·36-s − 0.308·42-s − 0.904·44-s + 0.577·48-s − 1.85·49-s + 1.41·50-s − 0.272·54-s − 1.05·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 627264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(627264\)    =    \(2^{6} \cdot 3^{4} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(39.9948\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 627264,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
3$C_1$ \( 1 + T \)
11$C_2$ \( 1 + 3 T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
7$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T^{2} ) \) 2.7.ab_o
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.17.a_d
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.23.a_ar
29$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.29.e_cj
31$C_2^2$ \( 1 + 47 T^{2} + p^{2} T^{4} \) 2.31.a_bv
37$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \) 2.37.a_abf
41$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \) 2.41.a_bf
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.a_cj
47$C_2^2$ \( 1 + 39 T^{2} + p^{2} T^{4} \) 2.47.a_bn
53$C_2^2$ \( 1 + 47 T^{2} + p^{2} T^{4} \) 2.53.a_bv
59$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.59.am_fx
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.c_cw
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.g_eo
71$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.71.a_w
73$C_2^2$ \( 1 - 99 T^{2} + p^{2} T^{4} \) 2.73.a_adv
79$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.k_ep
83$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \) 2.83.a_cj
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.ac_gw
97$C_2$$\times$$C_2$ \( ( 1 + 9 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.x_mi
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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

−8.082957146789492199335682439220, −7.55920326486073493580906458292, −7.07531406217681502369527855490, −6.67890882552647988105058048212, −6.22203214184557267430064738327, −5.58882272334205780678518308851, −5.43117895880557595096381860923, −4.85734304260639368778843915325, −4.55627878912994652701906282896, −3.96735278735811225785552498034, −3.37284126874933718948546043363, −2.82161173070628012203055023474, −2.23191669177233857694475067834, −1.36417429693726692119808261310, 0, 1.36417429693726692119808261310, 2.23191669177233857694475067834, 2.82161173070628012203055023474, 3.37284126874933718948546043363, 3.96735278735811225785552498034, 4.55627878912994652701906282896, 4.85734304260639368778843915325, 5.43117895880557595096381860923, 5.58882272334205780678518308851, 6.22203214184557267430064738327, 6.67890882552647988105058048212, 7.07531406217681502369527855490, 7.55920326486073493580906458292, 8.082957146789492199335682439220

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