Properties

Label 2-338130-1.1-c1-0-44
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s + 4·7-s − 8-s + 10-s + 3·11-s − 13-s − 4·14-s + 16-s + 19-s − 20-s − 3·22-s + 7·23-s + 25-s + 26-s + 4·28-s − 29-s + 3·31-s − 32-s − 4·35-s + 11·37-s − 38-s + 40-s − 4·41-s + 7·43-s + 3·44-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 + 0.904·11-s − 0.277·13-s − 1.06·14-s + 1/4·16-s + 0.229·19-s − 0.223·20-s − 0.639·22-s + 1.45·23-s + 1/5·25-s + 0.196·26-s + 0.755·28-s − 0.185·29-s + 0.538·31-s − 0.176·32-s − 0.676·35-s + 1.80·37-s − 0.162·38-s + 0.158·40-s − 0.624·41-s + 1.06·43-s + 0.452·44-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: \(0\)
Selberg data: \((2,\ 338130,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.054414958\)
\(L(\frac12)\) \(\approx\) \(3.054414958\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−12.36485944049876, −11.91242602483551, −11.67710888286860, −11.11561201649092, −10.84089244536549, −10.51917092081811, −9.624969337081318, −9.345427058229339, −8.998650498256131, −8.324205899532203, −8.063547684862470, −7.577767629449786, −7.227385858707758, −6.579083418639629, −6.252759105469462, −5.408446896821802, −5.078088792462315, −4.509899490066355, −4.051451368773467, −3.475338068615687, −2.680087924711892, −2.334017046255439, −1.481129093117967, −1.114958289863420, −0.5874014078943369, 0.5874014078943369, 1.114958289863420, 1.481129093117967, 2.334017046255439, 2.680087924711892, 3.475338068615687, 4.051451368773467, 4.509899490066355, 5.078088792462315, 5.408446896821802, 6.252759105469462, 6.579083418639629, 7.227385858707758, 7.577767629449786, 8.063547684862470, 8.324205899532203, 8.998650498256131, 9.345427058229339, 9.624969337081318, 10.51917092081811, 10.84089244536549, 11.11561201649092, 11.67710888286860, 11.91242602483551, 12.36485944049876

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