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Properties

Label 2-333270-1.1-c1-0-130
Degree $2$
Conductor $333270$
Sign $1$
Analytic cond. $2661.17$
Root an. cond. $51.5865$
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 + 7-s − 8-s + 10-s − 2·13-s − 14-s + 16-s − 2·17-s − 4·19-s − 20-s + 25-s + 2·26-s + 28-s − 4·29-s + 2·31-s − 32-s + 2·34-s − 35-s − 10·37-s + 4·38-s + 40-s + 6·41-s + 4·43-s − 10·47-s + 49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.223·20-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 0.742·29-s + 0.359·31-s − 0.176·32-s + 0.342·34-s − 0.169·35-s − 1.64·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s + 0.609·43-s − 1.45·47-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(333270\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(2661.17\)
Root analytic conductor: \(51.5865\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 333270,\ (\ :1/2),\ 1)\)

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 \( 1 + T \)
7 \( 1 - T \)
23 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
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

−13.02561769247891, −12.46554149971849, −12.02200808084452, −11.69752302676619, −11.10415607471258, −10.68453601369474, −10.53435465480973, −9.681047114254707, −9.486863216451934, −8.847651356703360, −8.464494899100614, −8.079683893705948, −7.517958191656474, −7.173022582158957, −6.650866491438485, −6.175581558244042, −5.598524341958879, −5.012428005685142, −4.486171931352978, −4.070509934715688, −3.360896050247609, −2.843737955323129, −2.196077432669538, −1.718801783505389, −1.101526601526115, 0, 0, 1.101526601526115, 1.718801783505389, 2.196077432669538, 2.843737955323129, 3.360896050247609, 4.070509934715688, 4.486171931352978, 5.012428005685142, 5.598524341958879, 6.175581558244042, 6.650866491438485, 7.173022582158957, 7.517958191656474, 8.079683893705948, 8.464494899100614, 8.847651356703360, 9.486863216451934, 9.681047114254707, 10.53435465480973, 10.68453601369474, 11.10415607471258, 11.69752302676619, 12.02200808084452, 12.46554149971849, 13.02561769247891

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