Properties

Label 2-338130-1.1-c1-0-60
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 − 6·11-s − 13-s + 2·14-s + 16-s + 5·19-s − 20-s + 6·22-s + 2·23-s + 25-s + 26-s − 2·28-s + 6·29-s + 2·31-s − 32-s + 2·35-s + 5·37-s − 5·38-s + 40-s + 3·41-s + 10·43-s − 6·44-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 − 1.80·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s + 1.14·19-s − 0.223·20-s + 1.27·22-s + 0.417·23-s + 1/5·25-s + 0.196·26-s − 0.377·28-s + 1.11·29-s + 0.359·31-s − 0.176·32-s + 0.338·35-s + 0.821·37-s − 0.811·38-s + 0.158·40-s + 0.468·41-s + 1.52·43-s − 0.904·44-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 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−12.76972100382017, −12.29202624924578, −11.94612749170994, −11.31485948664414, −10.91160923167702, −10.48440171268221, −10.05865503215024, −9.662837445100068, −9.220565853292409, −8.691713532139060, −8.074704858852202, −7.818532294125097, −7.411470590959686, −6.884351169877921, −6.437491809552975, −5.782057947594597, −5.299649829911929, −4.954784216066868, −4.174654288984041, −3.654017395583722, −2.886009219583015, −2.674469230003328, −2.253179718597151, −1.068428686650487, −0.7383336632173339, 0, 0.7383336632173339, 1.068428686650487, 2.253179718597151, 2.674469230003328, 2.886009219583015, 3.654017395583722, 4.174654288984041, 4.954784216066868, 5.299649829911929, 5.782057947594597, 6.437491809552975, 6.884351169877921, 7.411470590959686, 7.818532294125097, 8.074704858852202, 8.691713532139060, 9.220565853292409, 9.662837445100068, 10.05865503215024, 10.48440171268221, 10.91160923167702, 11.31485948664414, 11.94612749170994, 12.29202624924578, 12.76972100382017

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