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Properties

Label 2-404-404.355-c0-0-0
Degree $2$
Conductor $404$
Sign $0.650 + 0.759i$
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  + (0.535 − 0.844i)2-s + (−0.425 − 0.904i)4-s + (0.781 + 1.23i)5-s + (−0.992 − 0.125i)8-s + (0.876 − 0.481i)9-s + 1.45·10-s + (−1.80 + 0.713i)13-s + (−0.637 + 0.770i)16-s + (−0.5 − 1.53i)17-s + (0.0627 − 0.998i)18-s + (0.781 − 1.23i)20-s + (−0.479 + 1.01i)25-s + (−0.362 + 1.90i)26-s + (−0.574 + 0.227i)29-s + (0.309 + 0.951i)32-s + ⋯
L(s)  = 1  + (0.535 − 0.844i)2-s + (−0.425 − 0.904i)4-s + (0.781 + 1.23i)5-s + (−0.992 − 0.125i)8-s + (0.876 − 0.481i)9-s + 1.45·10-s + (−1.80 + 0.713i)13-s + (−0.637 + 0.770i)16-s + (−0.5 − 1.53i)17-s + (0.0627 − 0.998i)18-s + (0.781 − 1.23i)20-s + (−0.479 + 1.01i)25-s + (−0.362 + 1.90i)26-s + (−0.574 + 0.227i)29-s + (0.309 + 0.951i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(404\)    =    \(2^{2} \cdot 101\)
Sign: $0.650 + 0.759i$
Analytic conductor: \(0.201622\)
Root analytic conductor: \(0.449023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{404} (355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 404,\ (\ :0),\ 0.650 + 0.759i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.107020173\)
\(L(\frac12)\) \(\approx\) \(1.107020173\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.535 + 0.844i)T \)
101 \( 1 + (0.425 + 0.904i)T \)
good3 \( 1 + (-0.876 + 0.481i)T^{2} \)
5 \( 1 + (-0.781 - 1.23i)T + (-0.425 + 0.904i)T^{2} \)
7 \( 1 + (-0.728 - 0.684i)T^{2} \)
11 \( 1 + (-0.535 + 0.844i)T^{2} \)
13 \( 1 + (1.80 - 0.713i)T + (0.728 - 0.684i)T^{2} \)
17 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.187 - 0.982i)T^{2} \)
23 \( 1 + (0.992 - 0.125i)T^{2} \)
29 \( 1 + (0.574 - 0.227i)T + (0.728 - 0.684i)T^{2} \)
31 \( 1 + (-0.728 - 0.684i)T^{2} \)
37 \( 1 + (0.824 + 0.211i)T + (0.876 + 0.481i)T^{2} \)
41 \( 1 + (0.393 - 1.21i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + (0.929 + 0.368i)T^{2} \)
47 \( 1 + (0.929 - 0.368i)T^{2} \)
53 \( 1 + (-0.791 + 1.68i)T + (-0.637 - 0.770i)T^{2} \)
59 \( 1 + (0.187 + 0.982i)T^{2} \)
61 \( 1 + (-0.844 - 1.79i)T + (-0.637 + 0.770i)T^{2} \)
67 \( 1 + (-0.876 - 0.481i)T^{2} \)
71 \( 1 + (-0.876 + 0.481i)T^{2} \)
73 \( 1 + (0.0235 + 0.374i)T + (-0.992 + 0.125i)T^{2} \)
79 \( 1 + (0.992 + 0.125i)T^{2} \)
83 \( 1 + (0.992 + 0.125i)T^{2} \)
89 \( 1 + (1.11 + 1.35i)T + (-0.187 + 0.982i)T^{2} \)
97 \( 1 + (-0.791 - 1.68i)T + (-0.637 + 0.770i)T^{2} \)
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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

−11.45918966398805675967351543498, −10.37009739828409284640421543884, −9.786652814769247523098014659004, −9.213523081315331045028129083982, −7.13899156449858188749166543223, −6.75333301186423787084233134498, −5.38333251624530098551470077358, −4.36486075921149190783561283214, −2.95805389143499941705265659662, −2.07572310612402283037005324535, 2.12855828021913603191146722350, 4.05472181417673150820312765142, 5.01451889593210146500375003360, 5.61324785206970140102763491255, 6.90416673400715094744700455002, 7.83551156484182392742780071643, 8.714951585772870231831210486791, 9.640616630649771125144078187024, 10.46889060740650412024245699997, 12.17432513197770596071399633799

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