Properties

Label 2-404-404.283-c0-0-0
Degree $2$
Conductor $404$
Sign $0.675 + 0.737i$
Analytic cond. $0.201622$
Root an. cond. $0.449023$
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  + (0.728 − 0.684i)2-s + (0.0627 − 0.998i)4-s + (0.781 + 0.733i)5-s + (−0.637 − 0.770i)8-s + (−0.929 + 0.368i)9-s + 1.07·10-s + (−0.328 − 0.180i)13-s + (−0.992 − 0.125i)16-s + (−0.5 + 0.363i)17-s + (−0.425 + 0.904i)18-s + (0.781 − 0.733i)20-s + (0.00932 + 0.148i)25-s + (−0.362 + 0.0931i)26-s + (−1.41 − 0.779i)29-s + (−0.809 + 0.587i)32-s + ⋯
L(s)  = 1  + (0.728 − 0.684i)2-s + (0.0627 − 0.998i)4-s + (0.781 + 0.733i)5-s + (−0.637 − 0.770i)8-s + (−0.929 + 0.368i)9-s + 1.07·10-s + (−0.328 − 0.180i)13-s + (−0.992 − 0.125i)16-s + (−0.5 + 0.363i)17-s + (−0.425 + 0.904i)18-s + (0.781 − 0.733i)20-s + (0.00932 + 0.148i)25-s + (−0.362 + 0.0931i)26-s + (−1.41 − 0.779i)29-s + (−0.809 + 0.587i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(404\)    =    \(2^{2} \cdot 101\)
Sign: $0.675 + 0.737i$
Analytic conductor: \(0.201622\)
Root analytic conductor: \(0.449023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{404} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 404,\ (\ :0),\ 0.675 + 0.737i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.207470862\)
\(L(\frac12)\) \(\approx\) \(1.207470862\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.728 + 0.684i)T \)
101 \( 1 + (-0.0627 + 0.998i)T \)
good3 \( 1 + (0.929 - 0.368i)T^{2} \)
5 \( 1 + (-0.781 - 0.733i)T + (0.0627 + 0.998i)T^{2} \)
7 \( 1 + (-0.535 + 0.844i)T^{2} \)
11 \( 1 + (-0.728 + 0.684i)T^{2} \)
13 \( 1 + (0.328 + 0.180i)T + (0.535 + 0.844i)T^{2} \)
17 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.968 + 0.248i)T^{2} \)
23 \( 1 + (0.637 - 0.770i)T^{2} \)
29 \( 1 + (1.41 + 0.779i)T + (0.535 + 0.844i)T^{2} \)
31 \( 1 + (-0.535 + 0.844i)T^{2} \)
37 \( 1 + (0.0235 - 0.123i)T + (-0.929 - 0.368i)T^{2} \)
41 \( 1 + (-1.60 - 1.16i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 + (-0.876 + 0.481i)T^{2} \)
47 \( 1 + (-0.876 - 0.481i)T^{2} \)
53 \( 1 + (-0.110 - 1.74i)T + (-0.992 + 0.125i)T^{2} \)
59 \( 1 + (-0.968 - 0.248i)T^{2} \)
61 \( 1 + (0.0800 - 1.27i)T + (-0.992 - 0.125i)T^{2} \)
67 \( 1 + (0.929 + 0.368i)T^{2} \)
71 \( 1 + (0.929 - 0.368i)T^{2} \)
73 \( 1 + (0.824 + 1.75i)T + (-0.637 + 0.770i)T^{2} \)
79 \( 1 + (0.637 + 0.770i)T^{2} \)
83 \( 1 + (0.637 + 0.770i)T^{2} \)
89 \( 1 + (-1.84 + 0.233i)T + (0.968 - 0.248i)T^{2} \)
97 \( 1 + (-0.110 + 1.74i)T + (-0.992 - 0.125i)T^{2} \)
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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

−11.25749209511203387206908600300, −10.67110732039080522696554961294, −9.818955400658309282463512108626, −8.912070611387996128729526525677, −7.48521595213095589026946134790, −6.18934430949641370341460472237, −5.68831225960400433647677791144, −4.38921035350799099937578120352, −2.98532921749416349024449830239, −2.12865788153770767008274315388, 2.35699536337165054344293693292, 3.77571653576647881529836967252, 5.09665310514972593378497140383, 5.69180833476671623880333783064, 6.71346295789394127659764901614, 7.79620579704446194804503599628, 8.965504798062210882030332764742, 9.316743305182769298886486887278, 10.93188686156957912538798385189, 11.80604605846612985219955853762

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