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Properties

Label 2-404-404.239-c0-0-0
Degree $2$
Conductor $404$
Sign $0.909 + 0.415i$
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  + (−0.187 + 0.982i)2-s + (−0.929 − 0.368i)4-s + (−0.362 − 1.90i)5-s + (0.535 − 0.844i)8-s + (−0.637 − 0.770i)9-s + 1.93·10-s + (0.844 − 0.106i)13-s + (0.728 + 0.684i)16-s + (−0.5 + 0.363i)17-s + (0.876 − 0.481i)18-s + (−0.362 + 1.90i)20-s + (−2.55 + 1.01i)25-s + (−0.0534 + 0.849i)26-s + (1.60 − 0.202i)29-s + (−0.809 + 0.587i)32-s + ⋯
L(s)  = 1  + (−0.187 + 0.982i)2-s + (−0.929 − 0.368i)4-s + (−0.362 − 1.90i)5-s + (0.535 − 0.844i)8-s + (−0.637 − 0.770i)9-s + 1.93·10-s + (0.844 − 0.106i)13-s + (0.728 + 0.684i)16-s + (−0.5 + 0.363i)17-s + (0.876 − 0.481i)18-s + (−0.362 + 1.90i)20-s + (−2.55 + 1.01i)25-s + (−0.0534 + 0.849i)26-s + (1.60 − 0.202i)29-s + (−0.809 + 0.587i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(404\)    =    \(2^{2} \cdot 101\)
Sign: $0.909 + 0.415i$
Analytic conductor: \(0.201622\)
Root analytic conductor: \(0.449023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{404} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 404,\ (\ :0),\ 0.909 + 0.415i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6258329567\)
\(L(\frac12)\) \(\approx\) \(0.6258329567\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.187 - 0.982i)T \)
101 \( 1 + (0.929 + 0.368i)T \)
good3 \( 1 + (0.637 + 0.770i)T^{2} \)
5 \( 1 + (0.362 + 1.90i)T + (-0.929 + 0.368i)T^{2} \)
7 \( 1 + (-0.968 - 0.248i)T^{2} \)
11 \( 1 + (0.187 - 0.982i)T^{2} \)
13 \( 1 + (-0.844 + 0.106i)T + (0.968 - 0.248i)T^{2} \)
17 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.0627 + 0.998i)T^{2} \)
23 \( 1 + (-0.535 - 0.844i)T^{2} \)
29 \( 1 + (-1.60 + 0.202i)T + (0.968 - 0.248i)T^{2} \)
31 \( 1 + (-0.968 - 0.248i)T^{2} \)
37 \( 1 + (-0.791 - 1.68i)T + (-0.637 + 0.770i)T^{2} \)
41 \( 1 + (1.17 + 0.856i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 + (0.992 + 0.125i)T^{2} \)
47 \( 1 + (0.992 - 0.125i)T^{2} \)
53 \( 1 + (-1.84 + 0.730i)T + (0.728 - 0.684i)T^{2} \)
59 \( 1 + (-0.0627 - 0.998i)T^{2} \)
61 \( 1 + (0.996 + 0.394i)T + (0.728 + 0.684i)T^{2} \)
67 \( 1 + (0.637 - 0.770i)T^{2} \)
71 \( 1 + (0.637 + 0.770i)T^{2} \)
73 \( 1 + (-0.110 - 0.0604i)T + (0.535 + 0.844i)T^{2} \)
79 \( 1 + (-0.535 + 0.844i)T^{2} \)
83 \( 1 + (-0.535 + 0.844i)T^{2} \)
89 \( 1 + (0.929 - 0.872i)T + (0.0627 - 0.998i)T^{2} \)
97 \( 1 + (-1.84 - 0.730i)T + (0.728 + 0.684i)T^{2} \)
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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

−11.67632513723280884608813024602, −10.19460872653083352560989903694, −9.119284114263838995142902563221, −8.586200294402369596363571788650, −8.076894771306308114158974366662, −6.59647754748721000216870305386, −5.68722531415882950981399204060, −4.74925681672323304358952974105, −3.81852124290055734276165534742, −1.00930115635347560036163001700, 2.35680192243366114305958496144, 3.14092994858464580420160289037, 4.27311624138790197804338693670, 5.81921574722615473244405486055, 6.99832023618838197538436719024, 7.954286293541852403771852648961, 8.876002485851172333833648863643, 10.19530500832557369822779560473, 10.71154316668553788949698142542, 11.33721207839740917299584355515

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