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Properties

Label 2-455-455.139-c0-0-0
Degree $2$
Conductor $455$
Sign $-0.252 - 0.967i$
Analytic cond. $0.227074$
Root an. cond. $0.476523$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s − 5-s + (0.5 + 0.866i)7-s + (−1 + 1.73i)11-s + 0.999·12-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)15-s + (−0.499 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)20-s − 0.999·21-s + 25-s − 27-s + (0.499 − 0.866i)28-s + (0.5 − 0.866i)29-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s − 5-s + (0.5 + 0.866i)7-s + (−1 + 1.73i)11-s + 0.999·12-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)15-s + (−0.499 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)20-s − 0.999·21-s + 25-s − 27-s + (0.499 − 0.866i)28-s + (0.5 − 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(455\)    =    \(5 \cdot 7 \cdot 13\)
Sign: $-0.252 - 0.967i$
Analytic conductor: \(0.227074\)
Root analytic conductor: \(0.476523\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{455} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 455,\ (\ :0),\ -0.252 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4929103871\)
\(L(\frac12)\) \(\approx\) \(0.4929103871\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−11.38700202255182136235441639873, −10.67528969307244660997776893615, −9.808984638882259743834963486184, −9.093179579521636128665062956613, −8.012879652640960829789059354575, −6.92081481061817150377139060400, −5.52461500075370715233844099864, −4.69087798454991088710565208171, −4.32119370486387231448459076919, −2.19441434233136158123046028843, 0.71176571782114318137253800107, 3.18949321703132166199230157506, 4.00298385609636297940476810113, 5.30958642151659584887749989490, 6.55216891382531495509865238399, 7.60999899432040043504303058300, 8.096752793087174883545236170837, 8.756261215193763719426992660894, 10.61182942155104541292717807133, 11.06204359882601310163544332635

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