Properties

Label 2-338130-1.1-c1-0-30
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 − 11-s − 13-s + 2·14-s + 16-s − 5·19-s − 20-s + 22-s − 3·23-s + 25-s + 26-s − 2·28-s − 9·29-s − 3·31-s − 32-s + 2·35-s − 5·37-s + 5·38-s + 40-s + 8·41-s + 5·43-s − 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 − 0.301·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s − 1.14·19-s − 0.223·20-s + 0.213·22-s − 0.625·23-s + 1/5·25-s + 0.196·26-s − 0.377·28-s − 1.67·29-s − 0.538·31-s − 0.176·32-s + 0.338·35-s − 0.821·37-s + 0.811·38-s + 0.158·40-s + 1.24·41-s + 0.762·43-s − 0.150·44-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 \( 1 + T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−12.51112928173094, −12.42137443032475, −12.04028512548734, −11.15738919908247, −10.94948458182741, −10.63895438070521, −10.05555235851631, −9.489383552800632, −9.188866550386067, −8.842286518120537, −8.149293812853518, −7.666769285774014, −7.486916810355341, −6.840225461277833, −6.339892675604337, −5.909071200174022, −5.469825463289482, −4.683629841037758, −4.217593472184467, −3.597002140232356, −3.240532402479191, −2.437782555397273, −2.094584119161848, −1.400802679784014, −0.4704689461803507, 0, 0.4704689461803507, 1.400802679784014, 2.094584119161848, 2.437782555397273, 3.240532402479191, 3.597002140232356, 4.217593472184467, 4.683629841037758, 5.469825463289482, 5.909071200174022, 6.339892675604337, 6.840225461277833, 7.486916810355341, 7.666769285774014, 8.149293812853518, 8.842286518120537, 9.188866550386067, 9.489383552800632, 10.05555235851631, 10.63895438070521, 10.94948458182741, 11.15738919908247, 12.04028512548734, 12.42137443032475, 12.51112928173094

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