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Properties

Label 4-792e2-1.1-c1e2-0-122
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 + 2·4-s − 4·16-s + 4·17-s − 2·19-s − 6·25-s − 8·32-s + 8·34-s − 4·38-s − 12·41-s − 10·43-s − 10·49-s − 12·50-s − 8·64-s + 8·67-s + 8·68-s − 12·73-s − 4·76-s − 24·82-s − 16·83-s − 20·86-s + 2·89-s − 4·97-s − 20·98-s − 12·100-s + 30·113-s + 121-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 16-s + 0.970·17-s − 0.458·19-s − 6/5·25-s − 1.41·32-s + 1.37·34-s − 0.648·38-s − 1.87·41-s − 1.52·43-s − 1.42·49-s − 1.69·50-s − 64-s + 0.977·67-s + 0.970·68-s − 1.40·73-s − 0.458·76-s − 2.65·82-s − 1.75·83-s − 2.15·86-s + 0.211·89-s − 0.406·97-s − 2.02·98-s − 6/5·100-s + 2.82·113-s + 1/11·121-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: \(1\)
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 \( 1 \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ae_bi
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.c_o
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.23.a_o
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.29.a_g
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.37.a_ack
41$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.m_dy
43$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.k_ck
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.47.a_di
53$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.53.a_aba
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.61.a_abq
67$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.ai_ek
71$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \) 2.71.a_aes
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.79.a_afm
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.83.q_gk
89$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.89.ac_go
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.e_acg
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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

−8.044073724531568652575618525215, −7.71216341397497553733457081183, −6.99609020324212132882692258755, −6.72870277133947190736417516488, −6.23581126423677511509966818006, −5.64200819492977995161931488096, −5.49085577929950988360231315237, −4.73237147534793568242541887587, −4.55589233958700647673458918890, −3.74895625289394821675073785368, −3.42118066119039866629302067987, −2.97216790131392771832876343444, −2.11838496406239973085632915984, −1.54280325836205256397367872759, 0, 1.54280325836205256397367872759, 2.11838496406239973085632915984, 2.97216790131392771832876343444, 3.42118066119039866629302067987, 3.74895625289394821675073785368, 4.55589233958700647673458918890, 4.73237147534793568242541887587, 5.49085577929950988360231315237, 5.64200819492977995161931488096, 6.23581126423677511509966818006, 6.72870277133947190736417516488, 6.99609020324212132882692258755, 7.71216341397497553733457081183, 8.044073724531568652575618525215

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