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Properties

Label 2-40e2-80.69-c1-0-10
Degree $2$
Conductor $1600$
Sign $0.994 + 0.103i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.209 + 0.209i)3-s − 1.73·7-s − 2.91i·9-s + (−0.505 − 0.505i)11-s + (1.88 + 1.88i)13-s + 4.53i·17-s + (−3.22 + 3.22i)19-s + (−0.364 − 0.364i)21-s + 8.85·23-s + (1.23 − 1.23i)27-s + (2.44 − 2.44i)29-s + 5.70·31-s − 0.211i·33-s + (5.35 − 5.35i)37-s + 0.791i·39-s + ⋯
L(s)  = 1  + (0.120 + 0.120i)3-s − 0.656·7-s − 0.970i·9-s + (−0.152 − 0.152i)11-s + (0.523 + 0.523i)13-s + 1.09i·17-s + (−0.738 + 0.738i)19-s + (−0.0794 − 0.0794i)21-s + 1.84·23-s + (0.238 − 0.238i)27-s + (0.453 − 0.453i)29-s + 1.02·31-s − 0.0368i·33-s + (0.880 − 0.880i)37-s + 0.126i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.994 + 0.103i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.994 + 0.103i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.659561871\)
\(L(\frac12)\) \(\approx\) \(1.659561871\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.209 - 0.209i)T + 3iT^{2} \)
7 \( 1 + 1.73T + 7T^{2} \)
11 \( 1 + (0.505 + 0.505i)T + 11iT^{2} \)
13 \( 1 + (-1.88 - 1.88i)T + 13iT^{2} \)
17 \( 1 - 4.53iT - 17T^{2} \)
19 \( 1 + (3.22 - 3.22i)T - 19iT^{2} \)
23 \( 1 - 8.85T + 23T^{2} \)
29 \( 1 + (-2.44 + 2.44i)T - 29iT^{2} \)
31 \( 1 - 5.70T + 31T^{2} \)
37 \( 1 + (-5.35 + 5.35i)T - 37iT^{2} \)
41 \( 1 + 10.0iT - 41T^{2} \)
43 \( 1 + (-2.10 + 2.10i)T - 43iT^{2} \)
47 \( 1 + 4.32iT - 47T^{2} \)
53 \( 1 + (1.37 - 1.37i)T - 53iT^{2} \)
59 \( 1 + (-6.64 - 6.64i)T + 59iT^{2} \)
61 \( 1 + (-5.26 + 5.26i)T - 61iT^{2} \)
67 \( 1 + (-10.5 - 10.5i)T + 67iT^{2} \)
71 \( 1 + 14.0iT - 71T^{2} \)
73 \( 1 - 6.63T + 73T^{2} \)
79 \( 1 - 4.27T + 79T^{2} \)
83 \( 1 + (-9.15 - 9.15i)T + 83iT^{2} \)
89 \( 1 - 3.23iT - 89T^{2} \)
97 \( 1 - 1.94iT - 97T^{2} \)
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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

−9.258835019022998649849400540566, −8.765795712583636045238562324072, −7.923464669750575064003240595516, −6.67420203553139062855739681027, −6.39292567314372756745205822844, −5.38103120014044918276695797939, −4.07510735581807147952571159490, −3.57292562011931071779320026025, −2.38296581319483537050671749341, −0.878840754758189259831078364272, 0.951919907055188958175768299249, 2.58221145010091509040581665850, 3.12600971614765939349000824497, 4.63569648954771947656116123190, 5.11322327376968756289421575952, 6.37042611838999658513063266128, 6.93989109198897231163869051325, 7.903795636211173306184996420685, 8.555368834826548305186782337198, 9.465431935855835268661231536008

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