Properties

Label 2-338130-1.1-c1-0-87
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 4·7-s − 8-s + 10-s − 13-s + 4·14-s + 16-s − 2·19-s − 20-s − 4·23-s + 25-s + 26-s − 4·28-s − 2·29-s − 8·31-s − 32-s + 4·35-s + 4·37-s + 2·38-s + 40-s − 10·41-s − 10·43-s + 4·46-s + 9·49-s − 50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s − 0.353·8-s + 0.316·10-s − 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.458·19-s − 0.223·20-s − 0.834·23-s + 1/5·25-s + 0.196·26-s − 0.755·28-s − 0.371·29-s − 1.43·31-s − 0.176·32-s + 0.676·35-s + 0.657·37-s + 0.324·38-s + 0.158·40-s − 1.56·41-s − 1.52·43-s + 0.589·46-s + 9/7·49-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: \(2\)
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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−12.95725171322376, −12.50681282967674, −12.13187346342901, −11.66974028997861, −11.15820153053051, −10.69495040709951, −10.18754964461345, −9.805880289361381, −9.457172495296969, −9.026437324172596, −8.327942184846172, −8.190710793608137, −7.420917054015957, −7.089149015569027, −6.508513374473407, −6.351090667410320, −5.578949573956744, −5.193480037661688, −4.426004791367234, −3.713021752539301, −3.516412851883656, −2.940722026127040, −2.210859145237090, −1.814832319965650, −0.8944240289616788, 0, 0, 0.8944240289616788, 1.814832319965650, 2.210859145237090, 2.940722026127040, 3.516412851883656, 3.713021752539301, 4.426004791367234, 5.193480037661688, 5.578949573956744, 6.351090667410320, 6.508513374473407, 7.089149015569027, 7.420917054015957, 8.190710793608137, 8.327942184846172, 9.026437324172596, 9.457172495296969, 9.805880289361381, 10.18754964461345, 10.69495040709951, 11.15820153053051, 11.66974028997861, 12.13187346342901, 12.50681282967674, 12.95725171322376

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