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Properties

Label 4-588e2-1.1-c1e2-0-23
Degree $4$
Conductor $345744$
Sign $1$
Analytic cond. $22.0449$
Root an. cond. $2.16684$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 4·5-s + 3·8-s + 9-s − 4·10-s + 4·13-s − 16-s + 12·17-s − 18-s − 4·20-s + 2·25-s − 4·26-s − 4·29-s − 5·32-s − 12·34-s − 36-s + 12·37-s + 12·40-s − 4·41-s + 4·45-s − 2·50-s − 4·52-s + 12·53-s + 4·58-s + 4·61-s + 7·64-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.78·5-s + 1.06·8-s + 1/3·9-s − 1.26·10-s + 1.10·13-s − 1/4·16-s + 2.91·17-s − 0.235·18-s − 0.894·20-s + 2/5·25-s − 0.784·26-s − 0.742·29-s − 0.883·32-s − 2.05·34-s − 1/6·36-s + 1.97·37-s + 1.89·40-s − 0.624·41-s + 0.596·45-s − 0.282·50-s − 0.554·52-s + 1.64·53-s + 0.525·58-s + 0.512·61-s + 7/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(345744\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(22.0449\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 345744,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.970461553\)
\(L(\frac12)\) \(\approx\) \(1.970461553\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7 \( 1 \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.5.ae_o
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.37.am_eg
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.a_adu
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.89.abc_ok
97$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \) 2.97.bk_ty
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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

−9.090946623633278801475280262159, −8.050951011396219930904197989085, −7.977686193886306753143677629185, −7.70869243915887411656672354688, −6.69032425884932976329953320783, −6.53484921474790582970192984243, −5.58120289551734980246344429859, −5.48510765590071063887274763802, −5.36367026412859974981569612163, −4.01671863219499060051377207353, −3.99570051430579360707309966688, −3.06814457319249350690123364058, −2.26612547708831611492323080497, −1.40786771277750203137263322761, −1.10536343915846851689160557608, 1.10536343915846851689160557608, 1.40786771277750203137263322761, 2.26612547708831611492323080497, 3.06814457319249350690123364058, 3.99570051430579360707309966688, 4.01671863219499060051377207353, 5.36367026412859974981569612163, 5.48510765590071063887274763802, 5.58120289551734980246344429859, 6.53484921474790582970192984243, 6.69032425884932976329953320783, 7.70869243915887411656672354688, 7.977686193886306753143677629185, 8.050951011396219930904197989085, 9.090946623633278801475280262159

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