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Properties

Label 2-404-404.383-c0-0-0
Degree $2$
Conductor $404$
Sign $0.262 + 0.964i$
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  + (−0.637 − 0.770i)2-s + (−0.187 + 0.982i)4-s + (1.26 − 1.52i)5-s + (0.876 − 0.481i)8-s + (−0.425 + 0.904i)9-s − 1.98·10-s + (0.0672 + 1.06i)13-s + (−0.929 − 0.368i)16-s + (−0.5 − 1.53i)17-s + (0.968 − 0.248i)18-s + (1.26 + 1.52i)20-s + (−0.550 − 2.88i)25-s + (0.781 − 0.733i)26-s + (0.0388 + 0.616i)29-s + (0.309 + 0.951i)32-s + ⋯
L(s)  = 1  + (−0.637 − 0.770i)2-s + (−0.187 + 0.982i)4-s + (1.26 − 1.52i)5-s + (0.876 − 0.481i)8-s + (−0.425 + 0.904i)9-s − 1.98·10-s + (0.0672 + 1.06i)13-s + (−0.929 − 0.368i)16-s + (−0.5 − 1.53i)17-s + (0.968 − 0.248i)18-s + (1.26 + 1.52i)20-s + (−0.550 − 2.88i)25-s + (0.781 − 0.733i)26-s + (0.0388 + 0.616i)29-s + (0.309 + 0.951i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(404\)    =    \(2^{2} \cdot 101\)
Sign: $0.262 + 0.964i$
Analytic conductor: \(0.201622\)
Root analytic conductor: \(0.449023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{404} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 404,\ (\ :0),\ 0.262 + 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7096947754\)
\(L(\frac12)\) \(\approx\) \(0.7096947754\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.637 + 0.770i)T \)
101 \( 1 + (0.187 - 0.982i)T \)
good3 \( 1 + (0.425 - 0.904i)T^{2} \)
5 \( 1 + (-1.26 + 1.52i)T + (-0.187 - 0.982i)T^{2} \)
7 \( 1 + (0.992 + 0.125i)T^{2} \)
11 \( 1 + (0.637 + 0.770i)T^{2} \)
13 \( 1 + (-0.0672 - 1.06i)T + (-0.992 + 0.125i)T^{2} \)
17 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.728 + 0.684i)T^{2} \)
23 \( 1 + (-0.876 - 0.481i)T^{2} \)
29 \( 1 + (-0.0388 - 0.616i)T + (-0.992 + 0.125i)T^{2} \)
31 \( 1 + (0.992 + 0.125i)T^{2} \)
37 \( 1 + (0.200 - 0.316i)T + (-0.425 - 0.904i)T^{2} \)
41 \( 1 + (0.574 - 1.76i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + (-0.0627 + 0.998i)T^{2} \)
47 \( 1 + (-0.0627 - 0.998i)T^{2} \)
53 \( 1 + (0.0235 + 0.123i)T + (-0.929 + 0.368i)T^{2} \)
59 \( 1 + (-0.728 - 0.684i)T^{2} \)
61 \( 1 + (0.328 - 1.72i)T + (-0.929 - 0.368i)T^{2} \)
67 \( 1 + (0.425 + 0.904i)T^{2} \)
71 \( 1 + (0.425 - 0.904i)T^{2} \)
73 \( 1 + (-1.41 - 0.362i)T + (0.876 + 0.481i)T^{2} \)
79 \( 1 + (-0.876 + 0.481i)T^{2} \)
83 \( 1 + (-0.876 + 0.481i)T^{2} \)
89 \( 1 + (-0.791 + 0.313i)T + (0.728 - 0.684i)T^{2} \)
97 \( 1 + (0.0235 - 0.123i)T + (-0.929 - 0.368i)T^{2} \)
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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

−11.34377934122474879666261042079, −10.18455009842396069228816943279, −9.386917961502160364557923873316, −8.882705392441256805540436515853, −8.014052552646019312344371288757, −6.64036640394791228916777199794, −5.16157153845263846973103394147, −4.55295196068194995768215079145, −2.57495157628526989421961033027, −1.51516758300246311839125988170, 2.04627666451026924217809994555, 3.48135964939137236910383710553, 5.50376385215566355125683777431, 6.19448190551511404713360475305, 6.76347581415371052646798260035, 7.923302867835894405182343933571, 9.000525531577726466643737694163, 9.878621138732134609012998186161, 10.53937475786031084475241538855, 11.14476746815458238837157007793

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