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Properties

Label 4-2442e2-1.1-c1e2-0-21
Degree $4$
Conductor $5963364$
Sign $1$
Analytic cond. $380.229$
Root an. cond. $4.41582$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 4·5-s + 9-s + 16-s − 4·20-s − 10·23-s + 3·25-s − 12·31-s + 36-s − 3·37-s − 4·45-s + 4·47-s − 6·49-s − 13·53-s − 9·59-s + 64-s + 5·67-s − 5·71-s − 4·80-s + 81-s + 6·89-s − 10·92-s − 11·97-s + 3·100-s − 10·113-s + 40·115-s − 11·121-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.78·5-s + 1/3·9-s + 1/4·16-s − 0.894·20-s − 2.08·23-s + 3/5·25-s − 2.15·31-s + 1/6·36-s − 0.493·37-s − 0.596·45-s + 0.583·47-s − 6/7·49-s − 1.78·53-s − 1.17·59-s + 1/8·64-s + 0.610·67-s − 0.593·71-s − 0.447·80-s + 1/9·81-s + 0.635·89-s − 1.04·92-s − 1.11·97-s + 3/10·100-s − 0.940·113-s + 3.73·115-s − 121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5963364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5963364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5963364\)    =    \(2^{2} \cdot 3^{2} \cdot 11^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(380.229\)
Root analytic conductor: \(4.41582\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5963364,\ (\ :1/2, 1/2),\ 1)\)

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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 + p T^{2} \)
37$C_2$ \( 1 + 3 T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.e_n
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
13$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.13.a_u
17$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \) 2.17.a_p
19$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.19.a_f
23$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.k_cs
29$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \) 2.29.a_p
31$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.m_dt
41$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.41.a_m
43$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.43.a_z
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.ae_ac
53$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.n_fm
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.59.j_eo
61$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \) 2.61.a_dc
67$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.af_eq
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.71.f_fm
73$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \) 2.73.a_aef
79$C_2^2$ \( 1 + 85 T^{2} + p^{2} T^{4} \) 2.79.a_dh
83$C_2^2$ \( 1 + 99 T^{2} + p^{2} T^{4} \) 2.83.a_dv
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.ag_gw
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.97.l_gm
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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

−6.99571504092236802548854683223, −6.41242397050757051607678948889, −6.14917164914535400552370500657, −5.62545989747078753972046035447, −5.24245832001532259322742811300, −4.58076471920954526812924324952, −4.27716278845103445126029635429, −3.83173239778994022302064344666, −3.51201581242942984678218297360, −3.18586465218829135965508592426, −2.33575855295120087704049014582, −1.88169413049191545223872676782, −1.32597208483571942111295877695, 0, 0, 1.32597208483571942111295877695, 1.88169413049191545223872676782, 2.33575855295120087704049014582, 3.18586465218829135965508592426, 3.51201581242942984678218297360, 3.83173239778994022302064344666, 4.27716278845103445126029635429, 4.58076471920954526812924324952, 5.24245832001532259322742811300, 5.62545989747078753972046035447, 6.14917164914535400552370500657, 6.41242397050757051607678948889, 6.99571504092236802548854683223

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