Properties

Label 4-7098e2-1.1-c1e2-0-15
Degree $4$
Conductor $50381604$
Sign $1$
Analytic cond. $3212.37$
Root an. cond. $7.52846$
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  − 2·2-s − 2·3-s + 3·4-s + 2·5-s + 4·6-s + 2·7-s − 4·8-s + 3·9-s − 4·10-s + 4·11-s − 6·12-s − 4·14-s − 4·15-s + 5·16-s + 4·17-s − 6·18-s − 2·19-s + 6·20-s − 4·21-s − 8·22-s + 8·24-s − 4·25-s − 4·27-s + 6·28-s − 6·29-s + 8·30-s − 2·31-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s − 1.26·10-s + 1.20·11-s − 1.73·12-s − 1.06·14-s − 1.03·15-s + 5/4·16-s + 0.970·17-s − 1.41·18-s − 0.458·19-s + 1.34·20-s − 0.872·21-s − 1.70·22-s + 1.63·24-s − 4/5·25-s − 0.769·27-s + 1.13·28-s − 1.11·29-s + 1.46·30-s − 0.359·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50381604\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(3212.37\)
Root analytic conductor: \(7.52846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7098} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 50381604,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
13 \( 1 \)
good5$D_{4}$ \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 23 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 35 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 + 2 T - 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 80 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 2 T + 84 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 83 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 20 T + 219 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 6 T + 124 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 20 T + 234 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 175 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 2 T + 192 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−7.69675076313252080528249768866, −7.60986833057029388511277069466, −6.88953050141382994830620897328, −6.80402547409832091844043950583, −6.41140229713428440167108296644, −6.18263882204280117447801318066, −5.63207084629528928703412956674, −5.49089342120199712338979925913, −5.15917811862683093225710052693, −4.66841701760102615880274114316, −4.06289119058699427586216546193, −3.90439622504505069225325622869, −3.11830601020839149756811192277, −2.99156869565113052149832630402, −1.94362195251532692603271184588, −1.78446675843105315704759243920, −1.48947857188608562312616317028, −1.18380407588153173054292281309, 0, 0, 1.18380407588153173054292281309, 1.48947857188608562312616317028, 1.78446675843105315704759243920, 1.94362195251532692603271184588, 2.99156869565113052149832630402, 3.11830601020839149756811192277, 3.90439622504505069225325622869, 4.06289119058699427586216546193, 4.66841701760102615880274114316, 5.15917811862683093225710052693, 5.49089342120199712338979925913, 5.63207084629528928703412956674, 6.18263882204280117447801318066, 6.41140229713428440167108296644, 6.80402547409832091844043950583, 6.88953050141382994830620897328, 7.60986833057029388511277069466, 7.69675076313252080528249768866

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