Properties

Label 4-932261-1.1-c1e2-0-0
Degree $4$
Conductor $932261$
Sign $-1$
Analytic cond. $59.4417$
Root an. cond. $2.77666$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $3$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 4-s − 3·5-s + 2·6-s − 2·7-s + 6·10-s − 5·11-s − 12-s − 7·13-s + 4·14-s + 3·15-s + 16-s − 3·17-s − 4·19-s − 3·20-s + 2·21-s + 10·22-s − 5·23-s + 4·25-s + 14·26-s − 2·27-s − 2·28-s − 8·29-s − 6·30-s − 2·31-s + 2·32-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.816·6-s − 0.755·7-s + 1.89·10-s − 1.50·11-s − 0.288·12-s − 1.94·13-s + 1.06·14-s + 0.774·15-s + 1/4·16-s − 0.727·17-s − 0.917·19-s − 0.670·20-s + 0.436·21-s + 2.13·22-s − 1.04·23-s + 4/5·25-s + 2.74·26-s − 0.384·27-s − 0.377·28-s − 1.48·29-s − 1.09·30-s − 0.359·31-s + 0.353·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 932261 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 932261 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(932261\)    =    \(11 \cdot 84751\)
Sign: $-1$
Analytic conductor: \(59.4417\)
Root analytic conductor: \(2.77666\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{932261} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((4,\ 932261,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad11$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 4 T + p T^{2} ) \)
84751$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 395 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 7 T + 34 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 3 T - 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
31$D_{4}$ \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 28 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 2 T + 77 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
47$D_{4}$ \( 1 + 16 T + 129 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 22 T + 222 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$D_{4}$ \( 1 - 7 T + 118 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + T + 36 T^{2} + p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T + 24 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T - 68 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$D_{4}$ \( 1 + 8 T + 141 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−12.6452643602, −12.4869499204, −11.8157817204, −11.3869159100, −11.2425152499, −10.7895803027, −10.1789633370, −9.94029863130, −9.70454001350, −9.28812602093, −8.68087787459, −8.39516051734, −7.88060646239, −7.68136787710, −7.32123222978, −6.81580387927, −6.25216273626, −5.82999486213, −5.15658576735, −4.73390018136, −4.37082800633, −3.61048619948, −3.09640447209, −2.43647226108, −1.79406760087, 0, 0, 0, 1.79406760087, 2.43647226108, 3.09640447209, 3.61048619948, 4.37082800633, 4.73390018136, 5.15658576735, 5.82999486213, 6.25216273626, 6.81580387927, 7.32123222978, 7.68136787710, 7.88060646239, 8.39516051734, 8.68087787459, 9.28812602093, 9.70454001350, 9.94029863130, 10.1789633370, 10.7895803027, 11.2425152499, 11.3869159100, 11.8157817204, 12.4869499204, 12.6452643602

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