Properties

Label 4-15125-1.1-c1e2-0-3
Degree $4$
Conductor $15125$
Sign $-1$
Analytic cond. $0.964383$
Root an. cond. $0.990974$
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  − 3·4-s + 5-s − 6·9-s − 2·11-s + 5·16-s − 8·19-s − 3·20-s + 25-s + 12·29-s − 16·31-s + 18·36-s + 4·41-s + 6·44-s − 6·45-s − 14·49-s − 2·55-s + 8·59-s − 20·61-s − 3·64-s + 16·71-s + 24·76-s + 16·79-s + 5·80-s + 27·81-s + 20·89-s − 8·95-s + 12·99-s + ⋯
L(s)  = 1  − 3/2·4-s + 0.447·5-s − 2·9-s − 0.603·11-s + 5/4·16-s − 1.83·19-s − 0.670·20-s + 1/5·25-s + 2.22·29-s − 2.87·31-s + 3·36-s + 0.624·41-s + 0.904·44-s − 0.894·45-s − 2·49-s − 0.269·55-s + 1.04·59-s − 2.56·61-s − 3/8·64-s + 1.89·71-s + 2.75·76-s + 1.80·79-s + 0.559·80-s + 3·81-s + 2.11·89-s − 0.820·95-s + 1.20·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15125\)    =    \(5^{3} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(0.964383\)
Root analytic conductor: \(0.990974\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{15125} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 15125,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad5$C_1$ \( 1 - T \)
11$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
3$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−10.82193865307292862841462120688, −10.44930765910572775087302550356, −9.365267708430525132762162914792, −9.336297212975655357629362917616, −8.631003161371790257232477095324, −8.248642768475844318802620430882, −7.82885193609770565758085877155, −6.54918508631964327843106616026, −6.16821463973303244938343944630, −5.23353820210759730524176244525, −5.09766059609940450216711100862, −4.11816586584967359873614255585, −3.25423179226253980082861279896, −2.31804956482818418780108048277, 0, 2.31804956482818418780108048277, 3.25423179226253980082861279896, 4.11816586584967359873614255585, 5.09766059609940450216711100862, 5.23353820210759730524176244525, 6.16821463973303244938343944630, 6.54918508631964327843106616026, 7.82885193609770565758085877155, 8.248642768475844318802620430882, 8.631003161371790257232477095324, 9.336297212975655357629362917616, 9.365267708430525132762162914792, 10.44930765910572775087302550356, 10.82193865307292862841462120688

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