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Properties

Label 2-404-404.391-c0-0-0
Degree $2$
Conductor $404$
Sign $0.266 - 0.963i$
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  + (0.876 + 0.481i)2-s + (0.535 + 0.844i)4-s + (−1.62 + 0.895i)5-s + (0.0627 + 0.998i)8-s + (0.968 + 0.248i)9-s − 1.85·10-s + (0.371 − 1.94i)13-s + (−0.425 + 0.904i)16-s + (−0.5 + 0.363i)17-s + (0.728 + 0.684i)18-s + (−1.62 − 0.895i)20-s + (1.31 − 2.07i)25-s + (1.26 − 1.52i)26-s + (0.303 − 1.58i)29-s + (−0.809 + 0.587i)32-s + ⋯
L(s)  = 1  + (0.876 + 0.481i)2-s + (0.535 + 0.844i)4-s + (−1.62 + 0.895i)5-s + (0.0627 + 0.998i)8-s + (0.968 + 0.248i)9-s − 1.85·10-s + (0.371 − 1.94i)13-s + (−0.425 + 0.904i)16-s + (−0.5 + 0.363i)17-s + (0.728 + 0.684i)18-s + (−1.62 − 0.895i)20-s + (1.31 − 2.07i)25-s + (1.26 − 1.52i)26-s + (0.303 − 1.58i)29-s + (−0.809 + 0.587i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(404\)    =    \(2^{2} \cdot 101\)
Sign: $0.266 - 0.963i$
Analytic conductor: \(0.201622\)
Root analytic conductor: \(0.449023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{404} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 404,\ (\ :0),\ 0.266 - 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.115141029\)
\(L(\frac12)\) \(\approx\) \(1.115141029\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.876 - 0.481i)T \)
101 \( 1 + (-0.535 - 0.844i)T \)
good3 \( 1 + (-0.968 - 0.248i)T^{2} \)
5 \( 1 + (1.62 - 0.895i)T + (0.535 - 0.844i)T^{2} \)
7 \( 1 + (0.929 - 0.368i)T^{2} \)
11 \( 1 + (-0.876 - 0.481i)T^{2} \)
13 \( 1 + (-0.371 + 1.94i)T + (-0.929 - 0.368i)T^{2} \)
17 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.637 - 0.770i)T^{2} \)
23 \( 1 + (-0.0627 + 0.998i)T^{2} \)
29 \( 1 + (-0.303 + 1.58i)T + (-0.929 - 0.368i)T^{2} \)
31 \( 1 + (0.929 - 0.368i)T^{2} \)
37 \( 1 + (1.06 - 0.134i)T + (0.968 - 0.248i)T^{2} \)
41 \( 1 + (-0.688 - 0.500i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 + (0.187 + 0.982i)T^{2} \)
47 \( 1 + (0.187 - 0.982i)T^{2} \)
53 \( 1 + (0.200 - 0.316i)T + (-0.425 - 0.904i)T^{2} \)
59 \( 1 + (0.637 + 0.770i)T^{2} \)
61 \( 1 + (-0.0672 - 0.106i)T + (-0.425 + 0.904i)T^{2} \)
67 \( 1 + (-0.968 + 0.248i)T^{2} \)
71 \( 1 + (-0.968 - 0.248i)T^{2} \)
73 \( 1 + (0.929 - 0.872i)T + (0.0627 - 0.998i)T^{2} \)
79 \( 1 + (-0.0627 - 0.998i)T^{2} \)
83 \( 1 + (-0.0627 - 0.998i)T^{2} \)
89 \( 1 + (0.824 + 1.75i)T + (-0.637 + 0.770i)T^{2} \)
97 \( 1 + (0.200 + 0.316i)T + (-0.425 + 0.904i)T^{2} \)
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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

−11.74881890703525582755679497915, −10.93197892990552104677341674081, −10.26269268085724523640065374062, −8.307330681233399327469666476992, −7.80956347671629043466096606312, −7.06407759996346664812974838838, −6.04105645860618605559063321616, −4.63507153234087271409437330399, −3.78558349251806572454379672040, −2.85356446163337558246027193514, 1.53361378080098745944304798823, 3.57718833767186273170724443687, 4.30530615264840712755928236577, 4.96101602241353142066452480169, 6.70220235889685180642422297904, 7.27441964695970020092170709497, 8.698599450090189328988532489877, 9.428310866548496251284053152644, 10.80928903092427521320567539526, 11.52970463501815706213698176682

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