Properties

Label 4-8470e2-1.1-c1e2-0-22
Degree $4$
Conductor $71740900$
Sign $1$
Analytic cond. $4574.26$
Root an. cond. $8.22394$
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 − 6·12-s + 2·13-s + 4·14-s − 4·15-s + 5·16-s − 8·17-s − 6·18-s − 2·19-s + 6·20-s − 4·21-s − 2·23-s − 8·24-s + 3·25-s + 4·26-s + 14·27-s + 6·28-s − 8·30-s − 8·31-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 − 1.73·12-s + 0.554·13-s + 1.06·14-s − 1.03·15-s + 5/4·16-s − 1.94·17-s − 1.41·18-s − 0.458·19-s + 1.34·20-s − 0.872·21-s − 0.417·23-s − 1.63·24-s + 3/5·25-s + 0.784·26-s + 2.69·27-s + 1.13·28-s − 1.46·30-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(71740900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(4574.26\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 71740900,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T - 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T + 35 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 4 T - 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 8 T + 98 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 22 T + 267 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 28 T + 362 T^{2} + 28 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
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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

−7.27468816079358717287081324828, −7.06809249238730937872172131484, −6.66370393175572785690979127441, −6.50693757115073814119230721010, −5.97582395658598042744958914249, −5.91605449120081437454591989694, −5.44421919461885133098055542696, −5.37032755140582330383019732370, −4.86788744901633226904251558999, −4.57340714775145257396207789998, −4.24434621131384606767942713394, −3.84942939071688151868424208197, −3.26848108211266500882484582074, −2.88488677343228931508901998867, −2.45288131597304889879774458391, −2.16292706600134557001090543654, −1.47258214151679807390386033404, −1.36834154871966045404435928334, 0, 0, 1.36834154871966045404435928334, 1.47258214151679807390386033404, 2.16292706600134557001090543654, 2.45288131597304889879774458391, 2.88488677343228931508901998867, 3.26848108211266500882484582074, 3.84942939071688151868424208197, 4.24434621131384606767942713394, 4.57340714775145257396207789998, 4.86788744901633226904251558999, 5.37032755140582330383019732370, 5.44421919461885133098055542696, 5.91605449120081437454591989694, 5.97582395658598042744958914249, 6.50693757115073814119230721010, 6.66370393175572785690979127441, 7.06809249238730937872172131484, 7.27468816079358717287081324828

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