Properties

Label 4-6552e2-1.1-c1e2-0-10
Degree $4$
Conductor $42928704$
Sign $1$
Analytic cond. $2737.17$
Root an. cond. $7.23311$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s − 2·7-s + 2·11-s + 2·13-s − 6·17-s + 19-s − 5·23-s + 25-s − 5·29-s − 7·31-s − 6·35-s − 6·37-s + 4·41-s − 15·43-s − 15·47-s + 3·49-s − 53-s + 6·55-s + 16·59-s − 4·61-s + 6·65-s − 6·67-s + 4·71-s − 21·73-s − 4·77-s + 5·79-s + 25·83-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.755·7-s + 0.603·11-s + 0.554·13-s − 1.45·17-s + 0.229·19-s − 1.04·23-s + 1/5·25-s − 0.928·29-s − 1.25·31-s − 1.01·35-s − 0.986·37-s + 0.624·41-s − 2.28·43-s − 2.18·47-s + 3/7·49-s − 0.137·53-s + 0.809·55-s + 2.08·59-s − 0.512·61-s + 0.744·65-s − 0.733·67-s + 0.474·71-s − 2.45·73-s − 0.455·77-s + 0.562·79-s + 2.74·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42928704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42928704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(42928704\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(2737.17\)
Root analytic conductor: \(7.23311\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 42928704,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - T + 34 T^{2} - p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 7 T + 70 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 15 T + 138 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
47$C_4$ \( 1 + 15 T + 146 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + T + 68 T^{2} + p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 21 T + 252 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 5 T + 126 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 25 T + 318 T^{2} - 25 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - T + 72 T^{2} - p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + T + 156 T^{2} + p T^{3} + p^{2} T^{4} \)
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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

−7.76637239056065755281555838221, −7.52107256521758958207755911254, −6.85404975948450950797668240898, −6.69679172440800416587804732072, −6.42916898177105447712614891805, −6.23111723969295222987770984777, −5.58115999147749832034796371750, −5.55407839979654207077087147523, −5.05410699077309849918930569286, −4.67829370240888437049142254285, −3.96090910051573456201952195769, −3.89321305068040487314887289949, −3.41058722906170967680983244591, −3.03649761434942615382834913829, −2.21922679133019128242001569332, −2.19172977117506031332803069858, −1.57110979582852467478554247995, −1.33263030637445648412389089078, 0, 0, 1.33263030637445648412389089078, 1.57110979582852467478554247995, 2.19172977117506031332803069858, 2.21922679133019128242001569332, 3.03649761434942615382834913829, 3.41058722906170967680983244591, 3.89321305068040487314887289949, 3.96090910051573456201952195769, 4.67829370240888437049142254285, 5.05410699077309849918930569286, 5.55407839979654207077087147523, 5.58115999147749832034796371750, 6.23111723969295222987770984777, 6.42916898177105447712614891805, 6.69679172440800416587804732072, 6.85404975948450950797668240898, 7.52107256521758958207755911254, 7.76637239056065755281555838221

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