Properties

Label 2-338130-1.1-c1-0-73
Degree $2$
Conductor $338130$
Sign $-1$
Analytic cond. $2699.98$
Root an. cond. $51.9613$
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 + 4-s − 5-s − 8-s + 10-s + 11-s + 13-s + 16-s + 5·19-s − 20-s − 22-s + 23-s + 25-s − 26-s + 5·29-s + 7·31-s − 32-s + 3·37-s − 5·38-s + 40-s − 10·41-s + 7·43-s + 44-s − 46-s − 6·47-s − 7·49-s − 50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.277·13-s + 1/4·16-s + 1.14·19-s − 0.223·20-s − 0.213·22-s + 0.208·23-s + 1/5·25-s − 0.196·26-s + 0.928·29-s + 1.25·31-s − 0.176·32-s + 0.493·37-s − 0.811·38-s + 0.158·40-s − 1.56·41-s + 1.06·43-s + 0.150·44-s − 0.147·46-s − 0.875·47-s − 49-s − 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338130\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(2699.98\)
Root analytic conductor: \(51.9613\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{338130} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 338130,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 - T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−12.79835455864360, −12.15442500294911, −11.80137994532273, −11.38907843035026, −11.15129430641975, −10.31440464317898, −10.15895022541110, −9.603098466774242, −9.188868432470611, −8.576071015802383, −8.328267247685178, −7.691336730012503, −7.501119581806395, −6.680895210527312, −6.496546227766178, −5.981656108532398, −5.187229970313971, −4.861134644924136, −4.259532305816956, −3.576926321730158, −3.111067634892286, −2.707188894176279, −1.885238150232655, −1.246756050338871, −0.8211153550058275, 0, 0.8211153550058275, 1.246756050338871, 1.885238150232655, 2.707188894176279, 3.111067634892286, 3.576926321730158, 4.259532305816956, 4.861134644924136, 5.187229970313971, 5.981656108532398, 6.496546227766178, 6.680895210527312, 7.501119581806395, 7.691336730012503, 8.328267247685178, 8.576071015802383, 9.188868432470611, 9.603098466774242, 10.15895022541110, 10.31440464317898, 11.15129430641975, 11.38907843035026, 11.80137994532273, 12.15442500294911, 12.79835455864360

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