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Properties

Label 2-920-920.229-c0-0-8
Degree $2$
Conductor $920$
Sign $1$
Analytic cond. $0.459139$
Root an. cond. $0.677598$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.61·3-s + 4-s + 5-s − 1.61·6-s − 0.618·7-s − 8-s + 1.61·9-s − 10-s − 1.61·11-s + 1.61·12-s − 0.618·13-s + 0.618·14-s + 1.61·15-s + 16-s + 1.61·17-s − 1.61·18-s + 0.618·19-s + 20-s − 1.00·21-s + 1.61·22-s − 23-s − 1.61·24-s + 25-s + 0.618·26-s + 27-s − 0.618·28-s + ⋯
L(s)  = 1  − 2-s + 1.61·3-s + 4-s + 5-s − 1.61·6-s − 0.618·7-s − 8-s + 1.61·9-s − 10-s − 1.61·11-s + 1.61·12-s − 0.618·13-s + 0.618·14-s + 1.61·15-s + 16-s + 1.61·17-s − 1.61·18-s + 0.618·19-s + 20-s − 1.00·21-s + 1.61·22-s − 23-s − 1.61·24-s + 25-s + 0.618·26-s + 27-s − 0.618·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(0.459139\)
Root analytic conductor: \(0.677598\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (229, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.152845020\)
\(L(\frac12)\) \(\approx\) \(1.152845020\)
\(L(1)\) 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 \( 1 - T \)
23 \( 1 + T \)
good3 \( 1 - 1.61T + T^{2} \)
7 \( 1 + 0.618T + T^{2} \)
11 \( 1 + 1.61T + T^{2} \)
13 \( 1 + 0.618T + T^{2} \)
17 \( 1 - 1.61T + T^{2} \)
19 \( 1 - 0.618T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 0.618T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.61T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.61T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.61T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 1.61T + T^{2} \)
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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

−10.10764631094223678100554350910, −9.537024907993435210000574219111, −8.662226952428346057517387772224, −7.84260532296579166490641690562, −7.41627007879638152989695592363, −6.18340504695624293384765577469, −5.18599575100288424312082578909, −3.23939339767830745857496642205, −2.77196485177361156498303926706, −1.75237733296844512635455440748, 1.75237733296844512635455440748, 2.77196485177361156498303926706, 3.23939339767830745857496642205, 5.18599575100288424312082578909, 6.18340504695624293384765577469, 7.41627007879638152989695592363, 7.84260532296579166490641690562, 8.662226952428346057517387772224, 9.537024907993435210000574219111, 10.10764631094223678100554350910

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