Properties

Label 4-2340e2-1.1-c1e2-0-17
Degree $4$
Conductor $5475600$
Sign $1$
Analytic cond. $349.129$
Root an. cond. $4.32261$
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  − 2·5-s − 3·7-s + 3·11-s + 2·13-s − 7·17-s − 19-s − 7·23-s + 3·25-s − 5·29-s − 8·31-s + 6·35-s + 3·37-s + 7·41-s + 9·43-s − 16·47-s + 7·49-s + 12·53-s − 6·55-s + 5·59-s + 5·61-s − 4·65-s − 13·67-s − 3·71-s − 28·73-s − 9·77-s − 16·79-s − 24·83-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.13·7-s + 0.904·11-s + 0.554·13-s − 1.69·17-s − 0.229·19-s − 1.45·23-s + 3/5·25-s − 0.928·29-s − 1.43·31-s + 1.01·35-s + 0.493·37-s + 1.09·41-s + 1.37·43-s − 2.33·47-s + 49-s + 1.64·53-s − 0.809·55-s + 0.650·59-s + 0.640·61-s − 0.496·65-s − 1.58·67-s − 0.356·71-s − 3.27·73-s − 1.02·77-s − 1.80·79-s − 2.63·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5475600\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(349.129\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2340} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5475600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
13$C_2$ \( 1 - 2 T + p T^{2} \)
good7$C_2^2$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 7 T + 8 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 5 T - 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 11 T + 24 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
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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

−8.773453692928977125871023141085, −8.531230751751306782381720014366, −8.129896573527211396526231334783, −7.46544604244386318895949268125, −7.17183870093817072046984036586, −7.07333674843593360184764389978, −6.33642922060703740205905321251, −6.20429103168790069535178946288, −5.75314928937247293095406385081, −5.40212614971475008094060306723, −4.50173267951935701984822661979, −4.22562390836046947820236975264, −3.84311694788146426607907911327, −3.80261664299451090038676524863, −2.75771957710828053636665970441, −2.74699573493411003776391488007, −1.80747109841177323636992455344, −1.30256469810780121218017699756, 0, 0, 1.30256469810780121218017699756, 1.80747109841177323636992455344, 2.74699573493411003776391488007, 2.75771957710828053636665970441, 3.80261664299451090038676524863, 3.84311694788146426607907911327, 4.22562390836046947820236975264, 4.50173267951935701984822661979, 5.40212614971475008094060306723, 5.75314928937247293095406385081, 6.20429103168790069535178946288, 6.33642922060703740205905321251, 7.07333674843593360184764389978, 7.17183870093817072046984036586, 7.46544604244386318895949268125, 8.129896573527211396526231334783, 8.531230751751306782381720014366, 8.773453692928977125871023141085

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