Properties

Label 2-372810-1.1-c1-0-41
Degree $2$
Conductor $372810$
Sign $-1$
Analytic cond. $2976.90$
Root an. cond. $54.5610$
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 + 2·11-s − 12-s − 2·13-s − 2·14-s + 15-s + 16-s − 18-s − 6·19-s − 20-s − 2·21-s − 2·22-s − 6·23-s + 24-s + 25-s + 2·26-s − 27-s + 2·28-s + 10·29-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 + 0.603·11-s − 0.288·12-s − 0.554·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 1.37·19-s − 0.223·20-s − 0.436·21-s − 0.426·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.377·28-s + 1.85·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(372810\)    =    \(2 \cdot 3 \cdot 5 \cdot 17^{2} \cdot 43\)
Sign: $-1$
Analytic conductor: \(2976.90\)
Root analytic conductor: \(54.5610\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{372810} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 372810,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
17 \( 1 \)
43 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−12.40259710258709, −11.99313525277574, −11.81727710073044, −11.55341991559908, −10.73526676183607, −10.49584196486694, −10.09232706283896, −9.720700448900214, −8.948840164394199, −8.489395856456165, −8.279206765933455, −7.813291592379772, −7.172761588889993, −6.725513910655441, −6.408450634462311, −5.858546814387209, −5.274796110585578, −4.630839844055320, −4.272071993341316, −3.941325201541843, −2.904839019681104, −2.595246261373220, −1.767098173621177, −1.393305854177868, −0.6379361509627153, 0, 0.6379361509627153, 1.393305854177868, 1.767098173621177, 2.595246261373220, 2.904839019681104, 3.941325201541843, 4.272071993341316, 4.630839844055320, 5.274796110585578, 5.858546814387209, 6.408450634462311, 6.725513910655441, 7.172761588889993, 7.813291592379772, 8.279206765933455, 8.489395856456165, 8.948840164394199, 9.720700448900214, 10.09232706283896, 10.49584196486694, 10.73526676183607, 11.55341991559908, 11.81727710073044, 11.99313525277574, 12.40259710258709

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