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Properties

Label 2-920-920.229-c0-0-1
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 − 0.618·3-s + 4-s − 5-s + 0.618·6-s − 1.61·7-s − 8-s − 0.618·9-s + 10-s − 0.618·11-s − 0.618·12-s + 1.61·13-s + 1.61·14-s + 0.618·15-s + 16-s + 0.618·17-s + 0.618·18-s + 1.61·19-s − 20-s + 1.00·21-s + 0.618·22-s + 23-s + 0.618·24-s + 25-s − 1.61·26-s + 27-s − 1.61·28-s + ⋯
L(s)  = 1  − 2-s − 0.618·3-s + 4-s − 5-s + 0.618·6-s − 1.61·7-s − 8-s − 0.618·9-s + 10-s − 0.618·11-s − 0.618·12-s + 1.61·13-s + 1.61·14-s + 0.618·15-s + 16-s + 0.618·17-s + 0.618·18-s + 1.61·19-s − 20-s + 1.00·21-s + 0.618·22-s + 23-s + 0.618·24-s + 25-s − 1.61·26-s + 27-s − 1.61·28-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

\(L(\frac{1}{2})\) \(\approx\) \(0.3354517049\)
\(L(\frac12)\) \(\approx\) \(0.3354517049\)
\(L(1)\) 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 \( 1 + 0.618T + T^{2} \)
7 \( 1 + 1.61T + T^{2} \)
11 \( 1 + 0.618T + T^{2} \)
13 \( 1 - 1.61T + T^{2} \)
17 \( 1 - 0.618T + T^{2} \)
19 \( 1 - 1.61T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.61T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 0.618T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 0.618T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 0.618T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 0.618T + T^{2} \)
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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

−10.37041759899873819315775013523, −9.341748071578624560739884078387, −8.754580543074875853880029353581, −7.73868856757339279700430615696, −7.01857547602728152155274903369, −6.11184064382614756542755904250, −5.41980563912581369639956140715, −3.46639315980102355608248096477, −3.09623214243224520447086612842, −0.794954827623248637486156567358, 0.794954827623248637486156567358, 3.09623214243224520447086612842, 3.46639315980102355608248096477, 5.41980563912581369639956140715, 6.11184064382614756542755904250, 7.01857547602728152155274903369, 7.73868856757339279700430615696, 8.754580543074875853880029353581, 9.341748071578624560739884078387, 10.37041759899873819315775013523

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