Properties

Label 2-350064-1.1-c1-0-2
Degree $2$
Conductor $350064$
Sign $1$
Analytic cond. $2795.27$
Root an. cond. $52.8703$
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  + 2·5-s + 11-s + 13-s + 17-s − 8·19-s − 4·23-s − 25-s − 6·29-s + 4·31-s − 6·37-s + 6·41-s − 12·43-s + 8·47-s − 7·49-s − 2·53-s + 2·55-s + 4·59-s − 2·61-s + 2·65-s − 12·67-s + 8·71-s − 10·73-s − 8·79-s + 4·83-s + 2·85-s + 6·89-s − 16·95-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.301·11-s + 0.277·13-s + 0.242·17-s − 1.83·19-s − 0.834·23-s − 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.986·37-s + 0.937·41-s − 1.82·43-s + 1.16·47-s − 49-s − 0.274·53-s + 0.269·55-s + 0.520·59-s − 0.256·61-s + 0.248·65-s − 1.46·67-s + 0.949·71-s − 1.17·73-s − 0.900·79-s + 0.439·83-s + 0.216·85-s + 0.635·89-s − 1.64·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350064\)    =    \(2^{4} \cdot 3^{2} \cdot 11 \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(2795.27\)
Root analytic conductor: \(52.8703\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{350064} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 350064,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.182242991\)
\(L(\frac12)\) \(\approx\) \(1.182242991\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−12.62816872470322, −12.05941415838568, −11.72081104608930, −11.11287188258996, −10.71189953739061, −10.15027510392374, −9.975838609292246, −9.388384408128609, −8.831763613703999, −8.578216563880243, −8.001630880156807, −7.495008670027899, −6.911122922267065, −6.368681939719719, −6.046051572661594, −5.716159795171014, −4.997817282407369, −4.549391032000459, −3.913395506326532, −3.578476902671605, −2.785514739191851, −2.203311313612255, −1.773066679463805, −1.310416445711714, −0.2645749514608882, 0.2645749514608882, 1.310416445711714, 1.773066679463805, 2.203311313612255, 2.785514739191851, 3.578476902671605, 3.913395506326532, 4.549391032000459, 4.997817282407369, 5.716159795171014, 6.046051572661594, 6.368681939719719, 6.911122922267065, 7.495008670027899, 8.001630880156807, 8.578216563880243, 8.831763613703999, 9.388384408128609, 9.975838609292246, 10.15027510392374, 10.71189953739061, 11.11287188258996, 11.72081104608930, 12.05941415838568, 12.62816872470322

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