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Properties

Label 2-455-455.454-c0-0-6
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

\(L(\frac{1}{2})\) \(\approx\) \(1.385679690\)
\(L(\frac12)\) \(\approx\) \(1.385679690\)
\(L(1)\) 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

−11.53275670338738618225331340259, −10.26195380663044961028236744916, −9.167647061980475997668031966924, −8.497545394034329529082922726377, −7.926473478970942324410766675232, −6.96619615097941650788304395848, −5.09817636704050225698160834470, −4.30262930389533896931978006717, −3.50931168675912941907696705197, −2.28227882870049827896878400031, 2.28227882870049827896878400031, 3.50931168675912941907696705197, 4.30262930389533896931978006717, 5.09817636704050225698160834470, 6.96619615097941650788304395848, 7.926473478970942324410766675232, 8.497545394034329529082922726377, 9.167647061980475997668031966924, 10.26195380663044961028236744916, 11.53275670338738618225331340259

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