Properties

Label 4-8470e2-1.1-c1e2-0-23
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·3-s + 3·4-s + 2·5-s − 8·6-s − 2·7-s − 4·8-s + 6·9-s − 4·10-s + 12·12-s − 2·13-s + 4·14-s + 8·15-s + 5·16-s + 2·17-s − 12·18-s + 2·19-s + 6·20-s − 8·21-s − 2·23-s − 16·24-s + 3·25-s + 4·26-s − 4·27-s − 6·28-s − 6·29-s − 16·30-s + ⋯
L(s)  = 1  − 1.41·2-s + 2.30·3-s + 3/2·4-s + 0.894·5-s − 3.26·6-s − 0.755·7-s − 1.41·8-s + 2·9-s − 1.26·10-s + 3.46·12-s − 0.554·13-s + 1.06·14-s + 2.06·15-s + 5/4·16-s + 0.485·17-s − 2.82·18-s + 0.458·19-s + 1.34·20-s − 1.74·21-s − 0.417·23-s − 3.26·24-s + 3/5·25-s + 0.784·26-s − 0.769·27-s − 1.13·28-s − 1.11·29-s − 2.92·30-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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 14 T + 122 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T + 126 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 22 T + 282 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
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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

−7.59578488362350570607633471147, −7.58998736288240693963792346041, −7.24764137113673460524025099695, −6.79417802628564185442174553052, −6.34820821501692065303057537334, −6.00385949210381561133396615794, −5.58154166565594003828796728812, −5.53185000222495272857208147128, −4.60319927087410754035859961774, −4.46819384856622108442722786465, −3.56194860751061159554259077944, −3.43848688048706259786020070583, −3.00646838390008722974265510145, −2.98818382239517239835043381242, −2.24862362999248038185364216190, −2.11606209711189654783600413074, −1.49352115936949846850865989516, −1.41407636466437536620068272721, 0, 0, 1.41407636466437536620068272721, 1.49352115936949846850865989516, 2.11606209711189654783600413074, 2.24862362999248038185364216190, 2.98818382239517239835043381242, 3.00646838390008722974265510145, 3.43848688048706259786020070583, 3.56194860751061159554259077944, 4.46819384856622108442722786465, 4.60319927087410754035859961774, 5.53185000222495272857208147128, 5.58154166565594003828796728812, 6.00385949210381561133396615794, 6.34820821501692065303057537334, 6.79417802628564185442174553052, 7.24764137113673460524025099695, 7.58998736288240693963792346041, 7.59578488362350570607633471147

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