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Properties

Label 2-404-404.227-c0-0-0
Degree $2$
Conductor $404$
Sign $-0.454 - 0.890i$
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  + (0.0627 + 0.998i)2-s + (−0.992 + 0.125i)4-s + (−0.0534 + 0.849i)5-s + (−0.187 − 0.982i)8-s + (0.728 + 0.684i)9-s − 0.851·10-s + (−0.996 + 1.57i)13-s + (0.968 − 0.248i)16-s + (−0.5 − 1.53i)17-s + (−0.637 + 0.770i)18-s + (−0.0534 − 0.849i)20-s + (0.272 + 0.0344i)25-s + (−1.62 − 0.895i)26-s + (0.331 − 0.521i)29-s + (0.309 + 0.951i)32-s + ⋯
L(s)  = 1  + (0.0627 + 0.998i)2-s + (−0.992 + 0.125i)4-s + (−0.0534 + 0.849i)5-s + (−0.187 − 0.982i)8-s + (0.728 + 0.684i)9-s − 0.851·10-s + (−0.996 + 1.57i)13-s + (0.968 − 0.248i)16-s + (−0.5 − 1.53i)17-s + (−0.637 + 0.770i)18-s + (−0.0534 − 0.849i)20-s + (0.272 + 0.0344i)25-s + (−1.62 − 0.895i)26-s + (0.331 − 0.521i)29-s + (0.309 + 0.951i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(404\)    =    \(2^{2} \cdot 101\)
Sign: $-0.454 - 0.890i$
Analytic conductor: \(0.201622\)
Root analytic conductor: \(0.449023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{404} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 404,\ (\ :0),\ -0.454 - 0.890i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7711512526\)
\(L(\frac12)\) \(\approx\) \(0.7711512526\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0627 - 0.998i)T \)
101 \( 1 + (0.992 - 0.125i)T \)
good3 \( 1 + (-0.728 - 0.684i)T^{2} \)
5 \( 1 + (0.0534 - 0.849i)T + (-0.992 - 0.125i)T^{2} \)
7 \( 1 + (0.425 - 0.904i)T^{2} \)
11 \( 1 + (-0.0627 - 0.998i)T^{2} \)
13 \( 1 + (0.996 - 1.57i)T + (-0.425 - 0.904i)T^{2} \)
17 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.876 - 0.481i)T^{2} \)
23 \( 1 + (0.187 - 0.982i)T^{2} \)
29 \( 1 + (-0.331 + 0.521i)T + (-0.425 - 0.904i)T^{2} \)
31 \( 1 + (0.425 - 0.904i)T^{2} \)
37 \( 1 + (-1.84 + 0.730i)T + (0.728 - 0.684i)T^{2} \)
41 \( 1 + (-0.598 + 1.84i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + (-0.535 - 0.844i)T^{2} \)
47 \( 1 + (-0.535 + 0.844i)T^{2} \)
53 \( 1 + (1.06 + 0.134i)T + (0.968 + 0.248i)T^{2} \)
59 \( 1 + (-0.876 + 0.481i)T^{2} \)
61 \( 1 + (-0.371 + 0.0469i)T + (0.968 - 0.248i)T^{2} \)
67 \( 1 + (-0.728 + 0.684i)T^{2} \)
71 \( 1 + (-0.728 - 0.684i)T^{2} \)
73 \( 1 + (1.11 + 1.35i)T + (-0.187 + 0.982i)T^{2} \)
79 \( 1 + (0.187 + 0.982i)T^{2} \)
83 \( 1 + (0.187 + 0.982i)T^{2} \)
89 \( 1 + (-1.41 - 0.362i)T + (0.876 + 0.481i)T^{2} \)
97 \( 1 + (1.06 - 0.134i)T + (0.968 - 0.248i)T^{2} \)
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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

−11.78247382650335286484982218296, −10.79291805384254466919841728871, −9.673224012623154760268740296024, −9.136907398607323098114626560646, −7.63143182585497808570545347795, −7.17915174144493937278960405916, −6.37946602667772736202594321003, −4.94637818554111646552160520657, −4.22474258807621193690797168939, −2.50593712585465730763233365284, 1.22195378149233920644209065716, 2.90988851759274814253596789412, 4.19667598620394428233768589243, 5.01877413660447637873356552195, 6.22398414383354390072319894592, 7.84169724207102153709477143565, 8.561958071011961647411926312314, 9.647284491896537045874770787035, 10.19385038969567775863424310234, 11.19219033698420940120756278852

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