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Properties

Label 2-404-404.79-c0-0-0
Degree $2$
Conductor $404$
Sign $0.379 - 0.925i$
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  + (−0.425 + 0.904i)2-s + (−0.637 − 0.770i)4-s + (−0.0534 − 0.113i)5-s + (0.968 − 0.248i)8-s + (0.535 + 0.844i)9-s + 0.125·10-s + (1.27 + 1.19i)13-s + (−0.187 + 0.982i)16-s + (−0.5 − 0.363i)17-s + (−0.992 + 0.125i)18-s + (−0.0534 + 0.113i)20-s + (0.627 − 0.758i)25-s + (−1.62 + 0.645i)26-s + (−1.17 − 1.10i)29-s + (−0.809 − 0.587i)32-s + ⋯
L(s)  = 1  + (−0.425 + 0.904i)2-s + (−0.637 − 0.770i)4-s + (−0.0534 − 0.113i)5-s + (0.968 − 0.248i)8-s + (0.535 + 0.844i)9-s + 0.125·10-s + (1.27 + 1.19i)13-s + (−0.187 + 0.982i)16-s + (−0.5 − 0.363i)17-s + (−0.992 + 0.125i)18-s + (−0.0534 + 0.113i)20-s + (0.627 − 0.758i)25-s + (−1.62 + 0.645i)26-s + (−1.17 − 1.10i)29-s + (−0.809 − 0.587i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(404\)    =    \(2^{2} \cdot 101\)
Sign: $0.379 - 0.925i$
Analytic conductor: \(0.201622\)
Root analytic conductor: \(0.449023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{404} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 404,\ (\ :0),\ 0.379 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6797094691\)
\(L(\frac12)\) \(\approx\) \(0.6797094691\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.425 - 0.904i)T \)
101 \( 1 + (0.637 + 0.770i)T \)
good3 \( 1 + (-0.535 - 0.844i)T^{2} \)
5 \( 1 + (0.0534 + 0.113i)T + (-0.637 + 0.770i)T^{2} \)
7 \( 1 + (-0.0627 + 0.998i)T^{2} \)
11 \( 1 + (0.425 - 0.904i)T^{2} \)
13 \( 1 + (-1.27 - 1.19i)T + (0.0627 + 0.998i)T^{2} \)
17 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.929 - 0.368i)T^{2} \)
23 \( 1 + (-0.968 - 0.248i)T^{2} \)
29 \( 1 + (1.17 + 1.10i)T + (0.0627 + 0.998i)T^{2} \)
31 \( 1 + (-0.0627 + 0.998i)T^{2} \)
37 \( 1 + (1.11 - 0.614i)T + (0.535 - 0.844i)T^{2} \)
41 \( 1 + (-0.303 + 0.220i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 + (-0.728 + 0.684i)T^{2} \)
47 \( 1 + (-0.728 - 0.684i)T^{2} \)
53 \( 1 + (0.929 - 1.12i)T + (-0.187 - 0.982i)T^{2} \)
59 \( 1 + (0.929 + 0.368i)T^{2} \)
61 \( 1 + (1.23 + 1.49i)T + (-0.187 + 0.982i)T^{2} \)
67 \( 1 + (-0.535 + 0.844i)T^{2} \)
71 \( 1 + (-0.535 - 0.844i)T^{2} \)
73 \( 1 + (-1.84 - 0.233i)T + (0.968 + 0.248i)T^{2} \)
79 \( 1 + (-0.968 + 0.248i)T^{2} \)
83 \( 1 + (-0.968 + 0.248i)T^{2} \)
89 \( 1 + (0.200 + 1.05i)T + (-0.929 + 0.368i)T^{2} \)
97 \( 1 + (0.929 + 1.12i)T + (-0.187 + 0.982i)T^{2} \)
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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

−11.36422023317221214459233183831, −10.66325087122481618326769979148, −9.613449539863696340486919138778, −8.800614196328221215267030530845, −7.974977839508421395333161453730, −6.97129026281663843863809537794, −6.17434155540885942304115567804, −4.93564804112044017075576210384, −4.04620654365759893768252071202, −1.78528792672795120945510742226, 1.39312513326913642483377862286, 3.18118030037232536992208887046, 3.94089245229552878626317031829, 5.40390075168523663675415863508, 6.74511515439162489869860010909, 7.83105422754582868278210747195, 8.816644408693021056191282052391, 9.476138850048385574247420069905, 10.72289329207662105521000176766, 10.94219988388376934470589883101

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