Properties

Label 2-35490-1.1-c1-0-24
Degree $2$
Conductor $35490$
Sign $1$
Analytic cond. $283.389$
Root an. cond. $16.8341$
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 + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s − 4·11-s − 12-s + 14-s − 15-s + 16-s + 6·17-s + 18-s − 4·19-s + 20-s − 21-s − 4·22-s − 4·23-s − 24-s + 25-s − 27-s + 28-s + 2·29-s − 30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.218·21-s − 0.852·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.188·28-s + 0.371·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35490\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(283.389\)
Root analytic conductor: \(16.8341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{35490} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 35490,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.310315809\)
\(L(\frac12)\) \(\approx\) \(3.310315809\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.84989260291228, −14.37426036885146, −13.89004606803863, −13.30643486627119, −12.79675875716029, −12.41656340594952, −11.83807423742195, −11.33033775378828, −10.63642242296337, −10.31863780263350, −9.874828153537227, −9.099906191478235, −8.270602310192685, −7.687511622958231, −7.483581705389392, −6.344407559722784, −6.097370319056136, −5.580840432799741, −4.860227521373471, −4.534250269854263, −3.714632778983832, −2.869117607908310, −2.344369811070251, −1.511763679852404, −0.6290776851941945, 0.6290776851941945, 1.511763679852404, 2.344369811070251, 2.869117607908310, 3.714632778983832, 4.534250269854263, 4.860227521373471, 5.580840432799741, 6.097370319056136, 6.344407559722784, 7.483581705389392, 7.687511622958231, 8.270602310192685, 9.099906191478235, 9.874828153537227, 10.31863780263350, 10.63642242296337, 11.33033775378828, 11.83807423742195, 12.41656340594952, 12.79675875716029, 13.30643486627119, 13.89004606803863, 14.37426036885146, 14.84989260291228

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