Properties

Label 2-141120-1.1-c1-0-207
Degree $2$
Conductor $141120$
Sign $1$
Analytic cond. $1126.84$
Root an. cond. $33.5685$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 4·11-s − 2·13-s − 2·19-s + 4·23-s + 25-s + 10·29-s + 4·31-s + 2·37-s + 12·41-s + 4·43-s − 4·47-s + 2·53-s + 4·55-s + 10·59-s − 6·61-s − 2·65-s − 4·67-s + 12·71-s − 4·73-s − 4·79-s + 14·83-s − 8·89-s − 2·95-s − 8·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.20·11-s − 0.554·13-s − 0.458·19-s + 0.834·23-s + 1/5·25-s + 1.85·29-s + 0.718·31-s + 0.328·37-s + 1.87·41-s + 0.609·43-s − 0.583·47-s + 0.274·53-s + 0.539·55-s + 1.30·59-s − 0.768·61-s − 0.248·65-s − 0.488·67-s + 1.42·71-s − 0.468·73-s − 0.450·79-s + 1.53·83-s − 0.847·89-s − 0.205·95-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(141120\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1126.84\)
Root analytic conductor: \(33.5685\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{141120} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 141120,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.046369015\)
\(L(\frac12)\) \(\approx\) \(4.046369015\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
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

−13.38303995763218, −12.89669187292664, −12.42401876388279, −12.04549662801691, −11.47280125656907, −11.07317437059789, −10.44228176880312, −10.02901040913129, −9.539548776393940, −9.016180552301278, −8.705241399124739, −8.055325881194570, −7.489608667827174, −6.927517246528918, −6.359512707985212, −6.201586789455317, −5.391302846677051, −4.802062994168276, −4.363398530959575, −3.841279565203706, −2.996624862333998, −2.592122659579697, −1.928849533867471, −1.076905989800122, −0.7007844686777907, 0.7007844686777907, 1.076905989800122, 1.928849533867471, 2.592122659579697, 2.996624862333998, 3.841279565203706, 4.363398530959575, 4.802062994168276, 5.391302846677051, 6.201586789455317, 6.359512707985212, 6.927517246528918, 7.489608667827174, 8.055325881194570, 8.705241399124739, 9.016180552301278, 9.539548776393940, 10.02901040913129, 10.44228176880312, 11.07317437059789, 11.47280125656907, 12.04549662801691, 12.42401876388279, 12.89669187292664, 13.38303995763218

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