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Properties

Label 4-87e2-1.1-c1e2-0-0
Degree $4$
Conductor $7569$
Sign $1$
Analytic cond. $0.482606$
Root an. cond. $0.833485$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s − 2·4-s + 2·5-s + 2·6-s − 4·7-s − 3·8-s + 3·9-s + 2·10-s + 4·11-s − 4·12-s − 2·13-s − 4·14-s + 4·15-s + 16-s + 6·17-s + 3·18-s − 10·19-s − 4·20-s − 8·21-s + 4·22-s − 2·23-s − 6·24-s − 2·25-s − 2·26-s + 4·27-s + 8·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 4-s + 0.894·5-s + 0.816·6-s − 1.51·7-s − 1.06·8-s + 9-s + 0.632·10-s + 1.20·11-s − 1.15·12-s − 0.554·13-s − 1.06·14-s + 1.03·15-s + 1/4·16-s + 1.45·17-s + 0.707·18-s − 2.29·19-s − 0.894·20-s − 1.74·21-s + 0.852·22-s − 0.417·23-s − 1.22·24-s − 2/5·25-s − 0.392·26-s + 0.769·27-s + 1.51·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7569\)    =    \(3^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(0.482606\)
Root analytic conductor: \(0.833485\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7569,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.432343627\)
\(L(\frac12)\) \(\approx\) \(1.432343627\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 - T )^{2} \)
29$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 4 T + 53 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 146 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 30 T + 378 T^{2} + 30 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 6 T - 42 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−14.19656138291085195577105289556, −14.13784562908028887463274991736, −13.30822535943976405292619280405, −13.08126410111092776195400260092, −12.54842031959081348843701926982, −12.38690658068640418137094254295, −11.31442079433776982465479980330, −10.25919111520854853113468939749, −9.848199667372640298808001683822, −9.594911363765538106327790540763, −8.810338225621425421133394361693, −8.787088278780598681714873590880, −7.64544354559500736570238313155, −6.96227678862324707950148129167, −5.92447139966814564047583105421, −5.91703822599761461769955743526, −4.48583633468729403714404321989, −3.96740643999368951175333635084, −3.33284344106940796359913592368, −2.20009654735353763797367045973, 2.20009654735353763797367045973, 3.33284344106940796359913592368, 3.96740643999368951175333635084, 4.48583633468729403714404321989, 5.91703822599761461769955743526, 5.92447139966814564047583105421, 6.96227678862324707950148129167, 7.64544354559500736570238313155, 8.787088278780598681714873590880, 8.810338225621425421133394361693, 9.594911363765538106327790540763, 9.848199667372640298808001683822, 10.25919111520854853113468939749, 11.31442079433776982465479980330, 12.38690658068640418137094254295, 12.54842031959081348843701926982, 13.08126410111092776195400260092, 13.30822535943976405292619280405, 14.13784562908028887463274991736, 14.19656138291085195577105289556

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