Properties

Label 4-7098e2-1.1-c1e2-0-6
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 $0$

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 + 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 − 8·24-s − 4·25-s + 4·27-s + 6·28-s + 6·29-s − 8·30-s + 6·31-s − 6·32-s − 8·34-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 + 1.73·12-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 − 1.63·24-s − 4/5·25-s + 0.769·27-s + 1.13·28-s + 1.11·29-s − 1.46·30-s + 1.07·31-s − 1.06·32-s − 1.37·34-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: \(0\)
Selberg data: \((4,\ 50381604,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.461314208\)
\(L(\frac12)\) \(\approx\) \(4.461314208\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 35 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 55 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 10 T + 96 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 - 10 T + 108 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + 83 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 124 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 18 T + 191 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 10 T + 164 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 14 T + 215 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 14 T + 168 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
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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

−7.956631668692628380685886451970, −7.955666214384861745155888881305, −7.53492410206719796728550032373, −7.41917788607948480205610337193, −6.78796990123999126973912581581, −6.48620827009335767183281825138, −6.11487593894164456618605309450, −5.88245491672682120063772878323, −5.17973613290102546150606968292, −5.15918797883213657015177948409, −4.38255638184367424323185062122, −4.18096518372863589854625932736, −3.48246677510699934532439706511, −3.29594091665378146016179835872, −2.55833113192265060622001632738, −2.48554426471032434516161669766, −1.81979274375741587793014809330, −1.77865485860307062731330305620, −0.972442254934560421103330310643, −0.69385028253497057407204253047, 0.69385028253497057407204253047, 0.972442254934560421103330310643, 1.77865485860307062731330305620, 1.81979274375741587793014809330, 2.48554426471032434516161669766, 2.55833113192265060622001632738, 3.29594091665378146016179835872, 3.48246677510699934532439706511, 4.18096518372863589854625932736, 4.38255638184367424323185062122, 5.15918797883213657015177948409, 5.17973613290102546150606968292, 5.88245491672682120063772878323, 6.11487593894164456618605309450, 6.48620827009335767183281825138, 6.78796990123999126973912581581, 7.41917788607948480205610337193, 7.53492410206719796728550032373, 7.955666214384861745155888881305, 7.956631668692628380685886451970

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