Properties

Label 2-163254-1.1-c1-0-10
Degree $2$
Conductor $163254$
Sign $1$
Analytic cond. $1303.58$
Root an. cond. $36.1052$
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 + 3-s + 4-s − 3·5-s − 6-s + 7-s − 8-s + 9-s + 3·10-s − 4·11-s + 12-s − 14-s − 3·15-s + 16-s − 4·17-s − 18-s − 3·20-s + 21-s + 4·22-s + 23-s − 24-s + 4·25-s + 27-s + 28-s + 3·29-s + 3·30-s + 6·31-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 1.20·11-s + 0.288·12-s − 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.670·20-s + 0.218·21-s + 0.852·22-s + 0.208·23-s − 0.204·24-s + 4/5·25-s + 0.192·27-s + 0.188·28-s + 0.557·29-s + 0.547·30-s + 1.07·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(163254\)    =    \(2 \cdot 3 \cdot 7 \cdot 13^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(1303.58\)
Root analytic conductor: \(36.1052\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{163254} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 163254,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.218146194\)
\(L(\frac12)\) \(\approx\) \(1.218146194\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 \)
23 \( 1 - T \)
good5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−13.36201194063591, −12.67776903845369, −12.24222522776509, −11.77857412457977, −11.15191320736119, −11.05579739846920, −10.29112918649326, −10.03517259777904, −9.315960700846075, −8.786297591698659, −8.333353704004516, −7.989821528715901, −7.702087869265183, −7.120319190677728, −6.623759423884450, −6.068842563155174, −5.161369945155772, −4.743361536522325, −4.251319627695447, −3.586832207964909, −3.023014560682717, −2.471044001361910, −1.974258692047289, −0.9832821963482092, −0.4081034838659604, 0.4081034838659604, 0.9832821963482092, 1.974258692047289, 2.471044001361910, 3.023014560682717, 3.586832207964909, 4.251319627695447, 4.743361536522325, 5.161369945155772, 6.068842563155174, 6.623759423884450, 7.120319190677728, 7.702087869265183, 7.989821528715901, 8.333353704004516, 8.786297591698659, 9.315960700846075, 10.03517259777904, 10.29112918649326, 11.05579739846920, 11.15191320736119, 11.77857412457977, 12.24222522776509, 12.67776903845369, 13.36201194063591

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