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Properties

Label 4-1812608-1.1-c1e2-0-9
Degree $4$
Conductor $1812608$
Sign $-1$
Analytic cond. $115.573$
Root an. cond. $3.27879$
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-s + 4-s + 2·7-s − 8-s − 2·9-s + 6·11-s − 2·14-s + 16-s + 2·18-s − 6·22-s − 8·23-s − 6·25-s + 2·28-s − 2·29-s − 32-s − 2·36-s + 18·37-s + 6·44-s + 8·46-s − 3·49-s + 6·50-s − 16·53-s − 2·56-s + 2·58-s − 4·63-s + 64-s + 12·67-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 2/3·9-s + 1.80·11-s − 0.534·14-s + 1/4·16-s + 0.471·18-s − 1.27·22-s − 1.66·23-s − 6/5·25-s + 0.377·28-s − 0.371·29-s − 0.176·32-s − 1/3·36-s + 2.95·37-s + 0.904·44-s + 1.17·46-s − 3/7·49-s + 0.848·50-s − 2.19·53-s − 0.267·56-s + 0.262·58-s − 0.503·63-s + 1/8·64-s + 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1812608 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1812608 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1812608\)    =    \(2^{7} \cdot 7^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(115.573\)
Root analytic conductor: \(3.27879\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1812608,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
7$C_2$ \( 1 - 2 T + p T^{2} \)
17$C_2$ \( 1 + T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.11.ag_w
13$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.13.a_c
19$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.19.a_s
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.c_bi
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.31.a_o
37$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.37.as_fq
41$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.41.a_abq
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.47.a_ao
53$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.q_gk
59$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \) 2.59.a_co
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.61.a_ak
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.67.am_fe
71$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.i_fy
73$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.73.a_bi
79$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.79.a_abm
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.83.a_dm
89$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.89.a_be
97$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \) 2.97.a_fe
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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

−7.88392067883317873323732278829, −7.34193543734856141985766466012, −6.58921977174404859524933413249, −6.38156492095862524079412821342, −5.99634891770781982895329828519, −5.65283030338896677625816003056, −4.96614585117299175749938468359, −4.33704213063608854893331021225, −4.04456379579310693215902405013, −3.58134509166004526973746885075, −2.82970725166072879260547279980, −2.23795200362104456983294853979, −1.65256707523874384271938426178, −1.13607983509649299751343440314, 0, 1.13607983509649299751343440314, 1.65256707523874384271938426178, 2.23795200362104456983294853979, 2.82970725166072879260547279980, 3.58134509166004526973746885075, 4.04456379579310693215902405013, 4.33704213063608854893331021225, 4.96614585117299175749938468359, 5.65283030338896677625816003056, 5.99634891770781982895329828519, 6.38156492095862524079412821342, 6.58921977174404859524933413249, 7.34193543734856141985766466012, 7.88392067883317873323732278829

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