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Properties

Label 2-404-404.395-c0-0-0
Degree $2$
Conductor $404$
Sign $-0.204 + 0.978i$
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  + (−0.929 + 0.368i)2-s + (0.728 − 0.684i)4-s + (−1.62 − 0.645i)5-s + (−0.425 + 0.904i)8-s + (−0.187 − 0.982i)9-s + 1.75·10-s + (−1.23 − 0.317i)13-s + (0.0627 − 0.998i)16-s + (−0.5 − 1.53i)17-s + (0.535 + 0.844i)18-s + (−1.62 + 0.645i)20-s + (1.51 + 1.41i)25-s + (1.26 − 0.159i)26-s + (0.598 + 0.153i)29-s + (0.309 + 0.951i)32-s + ⋯
L(s)  = 1  + (−0.929 + 0.368i)2-s + (0.728 − 0.684i)4-s + (−1.62 − 0.645i)5-s + (−0.425 + 0.904i)8-s + (−0.187 − 0.982i)9-s + 1.75·10-s + (−1.23 − 0.317i)13-s + (0.0627 − 0.998i)16-s + (−0.5 − 1.53i)17-s + (0.535 + 0.844i)18-s + (−1.62 + 0.645i)20-s + (1.51 + 1.41i)25-s + (1.26 − 0.159i)26-s + (0.598 + 0.153i)29-s + (0.309 + 0.951i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(404\)    =    \(2^{2} \cdot 101\)
Sign: $-0.204 + 0.978i$
Analytic conductor: \(0.201622\)
Root analytic conductor: \(0.449023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{404} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 404,\ (\ :0),\ -0.204 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2910605349\)
\(L(\frac12)\) \(\approx\) \(0.2910605349\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.929 - 0.368i)T \)
101 \( 1 + (-0.728 + 0.684i)T \)
good3 \( 1 + (0.187 + 0.982i)T^{2} \)
5 \( 1 + (1.62 + 0.645i)T + (0.728 + 0.684i)T^{2} \)
7 \( 1 + (-0.876 + 0.481i)T^{2} \)
11 \( 1 + (0.929 - 0.368i)T^{2} \)
13 \( 1 + (1.23 + 0.317i)T + (0.876 + 0.481i)T^{2} \)
17 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.992 - 0.125i)T^{2} \)
23 \( 1 + (0.425 + 0.904i)T^{2} \)
29 \( 1 + (-0.598 - 0.153i)T + (0.876 + 0.481i)T^{2} \)
31 \( 1 + (-0.876 + 0.481i)T^{2} \)
37 \( 1 + (0.929 + 1.12i)T + (-0.187 + 0.982i)T^{2} \)
41 \( 1 + (-0.0388 + 0.119i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + (-0.968 + 0.248i)T^{2} \)
47 \( 1 + (-0.968 - 0.248i)T^{2} \)
53 \( 1 + (-1.41 - 1.32i)T + (0.0627 + 0.998i)T^{2} \)
59 \( 1 + (0.992 + 0.125i)T^{2} \)
61 \( 1 + (0.620 - 0.582i)T + (0.0627 - 0.998i)T^{2} \)
67 \( 1 + (0.187 - 0.982i)T^{2} \)
71 \( 1 + (0.187 + 0.982i)T^{2} \)
73 \( 1 + (1.06 - 1.67i)T + (-0.425 - 0.904i)T^{2} \)
79 \( 1 + (0.425 - 0.904i)T^{2} \)
83 \( 1 + (0.425 - 0.904i)T^{2} \)
89 \( 1 + (0.0235 + 0.374i)T + (-0.992 + 0.125i)T^{2} \)
97 \( 1 + (-1.41 + 1.32i)T + (0.0627 - 0.998i)T^{2} \)
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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

−11.34301023266642906856523002313, −10.16424933097510948451992387550, −9.100573041169892089651156127802, −8.606744516586332735180860203481, −7.37931239130538094015244434960, −7.10345500466573140491386260175, −5.47746504128092842234251683812, −4.38334771239013653799613487955, −2.89064656875672964790906512934, −0.51712563085711301580363819109, 2.28410289493462730775820183978, 3.52022946495997140160918868997, 4.62689925411822093833778709771, 6.52599579836020949537485693175, 7.42460956755769875198112934455, 8.038723400723256716961539226664, 8.793785329186530242709926015294, 10.28505798907262001489357149020, 10.66878377985526926998696825507, 11.67771004398062789160443097886

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