Properties

Label 2-374790-1.1-c1-0-56
Degree $2$
Conductor $374790$
Sign $-1$
Analytic cond. $2992.71$
Root an. cond. $54.7056$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s + 2·7-s + 8-s + 9-s − 10-s − 4·11-s + 12-s + 13-s + 2·14-s − 15-s + 16-s − 8·17-s + 18-s − 6·19-s − 20-s + 2·21-s − 4·22-s − 6·23-s + 24-s + 25-s + 26-s + 27-s + 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.277·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 1.94·17-s + 0.235·18-s − 1.37·19-s − 0.223·20-s + 0.436·21-s − 0.852·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.377·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(374790\)    =    \(2 \cdot 3 \cdot 5 \cdot 13 \cdot 31^{2}\)
Sign: $-1$
Analytic conductor: \(2992.71\)
Root analytic conductor: \(54.7056\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{374790} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 374790,\ (\ :1/2),\ -1)\)

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$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 - T \)
31 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.90348802474532, −12.38919837083781, −11.81519217382564, −11.30170872252796, −11.02925834670458, −10.57301274097888, −10.13575133689349, −9.648389360328411, −8.756168632765812, −8.558548859203895, −8.156748237383484, −7.835077284228965, −7.028754234715554, −6.863029979538796, −6.218540053831079, −5.689996291829926, −5.121042006831642, −4.572635951365377, −4.195520408284698, −3.955084244352945, −3.134586888321385, −2.473310484395371, −2.219191295678747, −1.773867216771024, −0.7513229242439229, 0, 0.7513229242439229, 1.773867216771024, 2.219191295678747, 2.473310484395371, 3.134586888321385, 3.955084244352945, 4.195520408284698, 4.572635951365377, 5.121042006831642, 5.689996291829926, 6.218540053831079, 6.863029979538796, 7.028754234715554, 7.835077284228965, 8.156748237383484, 8.558548859203895, 8.756168632765812, 9.648389360328411, 10.13575133689349, 10.57301274097888, 11.02925834670458, 11.30170872252796, 11.81519217382564, 12.38919837083781, 12.90348802474532

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