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Properties

Label 2-455-455.419-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: $\chi_{455} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 455,\ (\ :0),\ 0.252 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9638727428\)
\(L(\frac12)\) \(\approx\) \(0.9638727428\)
\(L(1)\) 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

−11.48908141224810735800235808833, −10.29743276344397234307744176434, −9.471241260632088447428551507705, −8.955999495979331236440036750930, −8.258591887702297113227171271029, −6.82759425896431698678252577377, −5.62584954094972502285689512295, −4.76741989474735286106793245275, −3.27480096911013775074372790387, −2.76302215069730645556988043706, 1.51400168849730338026252001844, 2.59758094519970056764261740654, 4.49615600777473905334390945698, 5.42576295839953750463627076591, 6.54131035477355808848413318198, 7.40381146627207671402324626920, 8.253664449046601831222489917644, 9.669584258580708780543620002116, 10.11113792584590306284107015707, 10.58228855874021703541133943961

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