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Properties

Label 2-176890-1.1-c1-0-57
Degree $2$
Conductor $176890$
Sign $1$
Analytic cond. $1412.47$
Root an. cond. $37.5828$
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 + 3·3-s + 4-s − 5-s + 3·6-s + 8-s + 6·9-s − 10-s − 5·11-s + 3·12-s + 5·13-s − 3·15-s + 16-s + 17-s + 6·18-s − 20-s − 5·22-s + 9·23-s + 3·24-s + 25-s + 5·26-s + 9·27-s − 5·29-s − 3·30-s + 7·31-s + 32-s − 15·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 1/2·4-s − 0.447·5-s + 1.22·6-s + 0.353·8-s + 2·9-s − 0.316·10-s − 1.50·11-s + 0.866·12-s + 1.38·13-s − 0.774·15-s + 1/4·16-s + 0.242·17-s + 1.41·18-s − 0.223·20-s − 1.06·22-s + 1.87·23-s + 0.612·24-s + 1/5·25-s + 0.980·26-s + 1.73·27-s − 0.928·29-s − 0.547·30-s + 1.25·31-s + 0.176·32-s − 2.61·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176890\)    =    \(2 \cdot 5 \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1412.47\)
Root analytic conductor: \(37.5828\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 176890,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(9.319947049\)
\(L(\frac12)\) \(\approx\) \(9.319947049\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 - p T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
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

−13.24000263772153, −12.95116040841234, −12.65197661205226, −11.63692063930486, −11.49155729003268, −10.62001492328141, −10.49018562920938, −9.900418036005762, −9.137324118055226, −8.799898395777259, −8.404900398028648, −7.815307298936895, −7.533966351144090, −7.078425332110032, −6.392767495426437, −5.809627417014525, −5.138788366139737, −4.618258436607781, −4.160790654503261, −3.417367096099077, −3.083799135132020, −2.858248253072500, −2.023890246764097, −1.473496454303632, −0.6788664414086619, 0.6788664414086619, 1.473496454303632, 2.023890246764097, 2.858248253072500, 3.083799135132020, 3.417367096099077, 4.160790654503261, 4.618258436607781, 5.138788366139737, 5.809627417014525, 6.392767495426437, 7.078425332110032, 7.533966351144090, 7.815307298936895, 8.404900398028648, 8.799898395777259, 9.137324118055226, 9.900418036005762, 10.49018562920938, 10.62001492328141, 11.49155729003268, 11.63692063930486, 12.65197661205226, 12.95116040841234, 13.24000263772153

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