Properties

Label 4-656e2-1.1-c1e2-0-20
Degree $4$
Conductor $430336$
Sign $1$
Analytic cond. $27.4385$
Root an. cond. $2.28870$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·5-s − 4·9-s + 6·23-s + 18·25-s − 6·31-s − 6·37-s + 2·41-s + 18·43-s − 24·45-s + 2·49-s + 14·59-s − 4·73-s + 7·81-s + 36·103-s − 12·107-s − 20·113-s + 36·115-s + 4·121-s + 30·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 36·155-s + 157-s + ⋯
L(s)  = 1  + 2.68·5-s − 4/3·9-s + 1.25·23-s + 18/5·25-s − 1.07·31-s − 0.986·37-s + 0.312·41-s + 2.74·43-s − 3.57·45-s + 2/7·49-s + 1.82·59-s − 0.468·73-s + 7/9·81-s + 3.54·103-s − 1.16·107-s − 1.88·113-s + 3.35·115-s + 4/11·121-s + 2.68·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 2.89·155-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(430336\)    =    \(2^{8} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(27.4385\)
Root analytic conductor: \(2.28870\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 430336,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.975906918\)
\(L(\frac12)\) \(\approx\) \(2.975906918\)
\(L(\frac{3}{2})\) 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 \( 1 \)
41$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
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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

−8.844217952915506412234055315954, −8.370913606857144041373699583631, −7.62293471903747372363922574204, −7.10127706958391854564286012609, −6.67468536408678964773964168044, −6.00796481812098980649780785362, −5.86009845593500951701617105668, −5.39048850508384291454270294704, −5.17953772462224133609964065964, −4.35688308262606674588223496560, −3.49957071718628007291438976681, −2.82745942726119673132939058271, −2.36049209002766580092855541607, −1.90717364128452830011571843750, −0.965933515453957879919128316287, 0.965933515453957879919128316287, 1.90717364128452830011571843750, 2.36049209002766580092855541607, 2.82745942726119673132939058271, 3.49957071718628007291438976681, 4.35688308262606674588223496560, 5.17953772462224133609964065964, 5.39048850508384291454270294704, 5.86009845593500951701617105668, 6.00796481812098980649780785362, 6.67468536408678964773964168044, 7.10127706958391854564286012609, 7.62293471903747372363922574204, 8.370913606857144041373699583631, 8.844217952915506412234055315954

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