Properties

Label 2-338130-1.1-c1-0-66
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 + 2·7-s − 8-s + 10-s + 4·11-s + 13-s − 2·14-s + 16-s − 4·19-s − 20-s − 4·22-s + 25-s − 26-s + 2·28-s + 6·29-s − 4·31-s − 32-s − 2·35-s + 4·38-s + 40-s − 6·41-s − 8·43-s + 4·44-s − 3·49-s − 50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s − 0.353·8-s + 0.316·10-s + 1.20·11-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.917·19-s − 0.223·20-s − 0.852·22-s + 1/5·25-s − 0.196·26-s + 0.377·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.338·35-s + 0.648·38-s + 0.158·40-s − 0.937·41-s − 1.21·43-s + 0.603·44-s − 3/7·49-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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
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

−12.68264949982431, −12.23453447675039, −11.72405280374497, −11.39361243473785, −11.12846888087500, −10.44709123724519, −10.17305462941693, −9.565123291611506, −9.061564985803133, −8.587553946872051, −8.341598023733584, −7.901386946390677, −7.291322162886728, −6.748839051102619, −6.498127037131137, −5.983829484382200, −5.158044783938795, −4.859827337692140, −4.158358711753223, −3.733317954206322, −3.238697908968300, −2.459102828500121, −1.852855984862599, −1.407190951424346, −0.7888370828283371, 0, 0.7888370828283371, 1.407190951424346, 1.852855984862599, 2.459102828500121, 3.238697908968300, 3.733317954206322, 4.158358711753223, 4.859827337692140, 5.158044783938795, 5.983829484382200, 6.498127037131137, 6.748839051102619, 7.291322162886728, 7.901386946390677, 8.341598023733584, 8.587553946872051, 9.061564985803133, 9.565123291611506, 10.17305462941693, 10.44709123724519, 11.12846888087500, 11.39361243473785, 11.72405280374497, 12.23453447675039, 12.68264949982431

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