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Properties

Label 4-396e2-1.1-c1e2-0-32
Degree $4$
Conductor $156816$
Sign $-1$
Analytic cond. $9.99872$
Root an. cond. $1.77822$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 3·8-s + 2·11-s − 4·13-s − 16-s + 2·22-s − 8·23-s + 6·25-s − 4·26-s + 5·32-s − 12·37-s − 2·44-s − 8·46-s + 16·47-s − 10·49-s + 6·50-s + 4·52-s − 8·59-s − 12·61-s + 7·64-s − 4·73-s − 12·74-s − 24·83-s − 6·88-s + 8·92-s + 16·94-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s + 0.603·11-s − 1.10·13-s − 1/4·16-s + 0.426·22-s − 1.66·23-s + 6/5·25-s − 0.784·26-s + 0.883·32-s − 1.97·37-s − 0.301·44-s − 1.17·46-s + 2.33·47-s − 1.42·49-s + 0.848·50-s + 0.554·52-s − 1.04·59-s − 1.53·61-s + 7/8·64-s − 0.468·73-s − 1.39·74-s − 2.63·83-s − 0.639·88-s + 0.834·92-s + 1.65·94-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(156816\)    =    \(2^{4} \cdot 3^{4} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(9.99872\)
Root analytic conductor: \(1.77822\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 156816,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - T + p T^{2} \)
3 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.43.a_by
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.61.m_gc
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.a_cs
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−8.847930541996233473124247216175, −8.765403563089383948261377682178, −8.057124954523707004822208987593, −7.53251376909507826007962870225, −6.96045911731745643617967557305, −6.52495869918911105195591443455, −5.82642873508535905813523337142, −5.52630761902370115381086518039, −4.77930282088861382836755069859, −4.44676656528528856043784213256, −3.85624161784864461163790617342, −3.18953257582755724401307416707, −2.56141659749378916924669768053, −1.55587956177828296160422082958, 0, 1.55587956177828296160422082958, 2.56141659749378916924669768053, 3.18953257582755724401307416707, 3.85624161784864461163790617342, 4.44676656528528856043784213256, 4.77930282088861382836755069859, 5.52630761902370115381086518039, 5.82642873508535905813523337142, 6.52495869918911105195591443455, 6.96045911731745643617967557305, 7.53251376909507826007962870225, 8.057124954523707004822208987593, 8.765403563089383948261377682178, 8.847930541996233473124247216175

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