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Properties

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

\(L(\frac{1}{2})\) \(\approx\) \(0.9381849501\)
\(L(\frac12)\) \(\approx\) \(0.9381849501\)
\(L(1)\) 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 \( 1 + 0.618T + T^{2} \)
3 \( 1 - 1.61T + T^{2} \)
11 \( 1 - T^{2} \)
17 \( 1 + 0.618T + T^{2} \)
19 \( 1 - 0.618T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.61T + T^{2} \)
31 \( 1 + 1.61T + T^{2} \)
37 \( 1 - 1.61T + T^{2} \)
41 \( 1 - 0.618T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.61T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 0.618T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 0.618T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.61T + T^{2} \)
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

−10.82977439119413012017552195011, −9.754763886671394839970612738405, −9.411501007949812269267871737463, −9.005863458718882944939497584632, −7.78805038284386544508188525978, −7.10824910606955389876220228778, −5.63466983010195569256531302668, −4.25774802003994057662337402723, −3.06335096704714708565670924948, −1.95218191586003584682661967050, 1.95218191586003584682661967050, 3.06335096704714708565670924948, 4.25774802003994057662337402723, 5.63466983010195569256531302668, 7.10824910606955389876220228778, 7.78805038284386544508188525978, 9.005863458718882944939497584632, 9.411501007949812269267871737463, 9.754763886671394839970612738405, 10.82977439119413012017552195011

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