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Properties

Label 4-408e2-1.1-c1e2-0-58
Degree $4$
Conductor $166464$
Sign $-1$
Analytic cond. $10.6138$
Root an. cond. $1.80496$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 4·5-s − 3·8-s − 3·9-s + 4·10-s − 16-s − 3·18-s − 8·19-s − 4·20-s − 8·23-s + 2·25-s − 12·29-s + 5·32-s + 3·36-s − 8·38-s − 12·40-s + 8·43-s − 12·45-s − 8·46-s + 2·49-s + 2·50-s − 12·53-s − 12·58-s + 7·64-s + 8·67-s + 8·71-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 1.78·5-s − 1.06·8-s − 9-s + 1.26·10-s − 1/4·16-s − 0.707·18-s − 1.83·19-s − 0.894·20-s − 1.66·23-s + 2/5·25-s − 2.22·29-s + 0.883·32-s + 1/2·36-s − 1.29·38-s − 1.89·40-s + 1.21·43-s − 1.78·45-s − 1.17·46-s + 2/7·49-s + 0.282·50-s − 1.64·53-s − 1.57·58-s + 7/8·64-s + 0.977·67-s + 0.949·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(166464\)    =    \(2^{6} \cdot 3^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(10.6138\)
Root analytic conductor: \(1.80496\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 166464,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.5.ae_o
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.71.ai_gc
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.79.a_o
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
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

−8.872689562844898968412116321231, −8.868215889831003694053677970209, −7.86052886694471390828228448204, −7.81886564473676875900189501384, −6.64458936246770473314630815851, −6.12380991478521668699719766019, −6.04868829414574942081046284670, −5.50246131433375739774113763395, −5.15238580447119340700000765704, −4.18976898855068708054002343552, −3.94486422776053801827158365621, −3.06428895368232173582353504603, −2.14503914986953664301144200995, −1.98635588682742520402514989313, 0, 1.98635588682742520402514989313, 2.14503914986953664301144200995, 3.06428895368232173582353504603, 3.94486422776053801827158365621, 4.18976898855068708054002343552, 5.15238580447119340700000765704, 5.50246131433375739774113763395, 6.04868829414574942081046284670, 6.12380991478521668699719766019, 6.64458936246770473314630815851, 7.81886564473676875900189501384, 7.86052886694471390828228448204, 8.868215889831003694053677970209, 8.872689562844898968412116321231

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