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Properties

Label 4-792e2-1.1-c1e2-0-17
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 3·8-s − 16-s − 2·25-s − 12·29-s + 12·31-s + 5·32-s + 12·37-s − 4·41-s + 6·49-s − 2·50-s − 12·58-s + 12·62-s + 7·64-s + 4·67-s + 12·74-s − 4·82-s − 8·83-s − 12·97-s + 6·98-s + 2·100-s − 8·101-s − 12·103-s + 8·107-s + 12·116-s − 11·121-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 1/4·16-s − 2/5·25-s − 2.22·29-s + 2.15·31-s + 0.883·32-s + 1.97·37-s − 0.624·41-s + 6/7·49-s − 0.282·50-s − 1.57·58-s + 1.52·62-s + 7/8·64-s + 0.488·67-s + 1.39·74-s − 0.441·82-s − 0.878·83-s − 1.21·97-s + 0.606·98-s + 1/5·100-s − 0.796·101-s − 1.18·103-s + 0.773·107-s + 1.11·116-s − 121-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: \(0\)
Selberg data: \((4,\ 627264,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.837933184\)
\(L(\frac12)\) \(\approx\) \(1.837933184\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
3 \( 1 \)
11$C_2$ \( 1 + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.13.a_ag
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.19.a_ag
23$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.23.a_ag
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.31.am_dq
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.am_dq
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.e_w
43$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.43.a_abe
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.53.a_cg
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.59.a_ak
61$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.61.a_s
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.ae_dy
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.71.a_bi
73$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \) 2.73.a_aek
79$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.79.a_s
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.i_eo
89$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.89.a_abe
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.m_ig
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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

−8.257835717554447702509996377195, −7.966843143564028815015951200767, −7.61824169768911989600687753248, −6.85713763099235549236560096642, −6.51350898621945894029384095171, −6.00608168727567737303919926617, −5.43840266214952900746837752147, −5.32481407480352457411323412128, −4.41855202323956480908532485126, −4.22553319983239249938991147891, −3.75494768892754773946124087377, −2.95816448772345611225967462056, −2.61948483359941673148206408677, −1.67704347899440921477501835861, −0.62162056746197416242969472493, 0.62162056746197416242969472493, 1.67704347899440921477501835861, 2.61948483359941673148206408677, 2.95816448772345611225967462056, 3.75494768892754773946124087377, 4.22553319983239249938991147891, 4.41855202323956480908532485126, 5.32481407480352457411323412128, 5.43840266214952900746837752147, 6.00608168727567737303919926617, 6.51350898621945894029384095171, 6.85713763099235549236560096642, 7.61824169768911989600687753248, 7.966843143564028815015951200767, 8.257835717554447702509996377195

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