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Properties

Label 2-455-455.454-c0-0-7
Degree $2$
Conductor $455$
Sign $1$
Analytic cond. $0.227074$
Root an. cond. $0.476523$
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  + 1.61·2-s − 0.618·3-s + 1.61·4-s + 5-s − 1.00·6-s − 7-s + 8-s − 0.618·9-s + 1.61·10-s − 1.00·12-s − 13-s − 1.61·14-s − 0.618·15-s + 1.61·17-s − 0.999·18-s − 1.61·19-s + 1.61·20-s + 0.618·21-s − 0.618·24-s + 25-s − 1.61·26-s + 27-s − 1.61·28-s + 0.618·29-s − 1.00·30-s + 0.618·31-s − 32-s + ⋯
L(s)  = 1  + 1.61·2-s − 0.618·3-s + 1.61·4-s + 5-s − 1.00·6-s − 7-s + 8-s − 0.618·9-s + 1.61·10-s − 1.00·12-s − 13-s − 1.61·14-s − 0.618·15-s + 1.61·17-s − 0.999·18-s − 1.61·19-s + 1.61·20-s + 0.618·21-s − 0.618·24-s + 25-s − 1.61·26-s + 27-s − 1.61·28-s + 0.618·29-s − 1.00·30-s + 0.618·31-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(455\)    =    \(5 \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(0.227074\)
Root analytic conductor: \(0.476523\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{455} (454, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 455,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.604158131\)
\(L(\frac12)\) \(\approx\) \(1.604158131\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 + T \)
good2 \( 1 - 1.61T + T^{2} \)
3 \( 1 + 0.618T + T^{2} \)
11 \( 1 - T^{2} \)
17 \( 1 - 1.61T + T^{2} \)
19 \( 1 + 1.61T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 0.618T + T^{2} \)
31 \( 1 - 0.618T + T^{2} \)
37 \( 1 + 0.618T + T^{2} \)
41 \( 1 + 1.61T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 0.618T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 1.61T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 1.61T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−11.70069660327717277702905097054, −10.47069501557277608534698147222, −9.903313649497054664411617526642, −8.609445141384698874747377257719, −6.92559653171325574980040102120, −6.27172015635440273509783693775, −5.56472105591527686146600246860, −4.81203407246921523038393131124, −3.36642887434298895902636875765, −2.42164990991445904924649273207, 2.42164990991445904924649273207, 3.36642887434298895902636875765, 4.81203407246921523038393131124, 5.56472105591527686146600246860, 6.27172015635440273509783693775, 6.92559653171325574980040102120, 8.609445141384698874747377257719, 9.903313649497054664411617526642, 10.47069501557277608534698147222, 11.70069660327717277702905097054

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