Properties

Label 2-40e2-8.5-c1-0-33
Degree $2$
Conductor $1600$
Sign $-0.258 + 0.965i$
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  + 0.732i·3-s − 1.26·7-s + 2.46·9-s − 3.46i·11-s − 3.46i·13-s − 3.46·17-s − 2i·19-s − 0.928i·21-s − 8.19·23-s + 4i·27-s − 9.46·31-s + 2.53·33-s + 6i·37-s + 2.53·39-s + 2.53·41-s + ⋯
L(s)  = 1  + 0.422i·3-s − 0.479·7-s + 0.821·9-s − 1.04i·11-s − 0.960i·13-s − 0.840·17-s − 0.458i·19-s − 0.202i·21-s − 1.70·23-s + 0.769i·27-s − 1.69·31-s + 0.441·33-s + 0.986i·37-s + 0.406·39-s + 0.396·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.258 + 0.965i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.258 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9267251933\)
\(L(\frac12)\) \(\approx\) \(0.9267251933\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.732iT - 3T^{2} \)
7 \( 1 + 1.26T + 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + 8.19T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 9.46T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 2.53T + 41T^{2} \)
43 \( 1 + 10.1iT - 43T^{2} \)
47 \( 1 - 8.19T + 47T^{2} \)
53 \( 1 + 10.3iT - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + 12.9iT - 61T^{2} \)
67 \( 1 + 10.1iT - 67T^{2} \)
71 \( 1 + 4.39T + 71T^{2} \)
73 \( 1 - 14.3T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + 4.73iT - 83T^{2} \)
89 \( 1 - 0.928T + 89T^{2} \)
97 \( 1 - 6.39T + 97T^{2} \)
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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

−9.236016914716632101969488478226, −8.455600646678025268710941267333, −7.62064198156454184464574209902, −6.71232668698640180797667246562, −5.89347622653997201109091335189, −5.05734636538964618529694229974, −3.95983321760789105428529623148, −3.32242106923152186483872695255, −2.01353434820075179885798404821, −0.34267446449125540644934033319, 1.60447023758483719775756512531, 2.36071409947482893284312822335, 3.99733779439817723181855403385, 4.36156406636242263863830466630, 5.72495468034720360008856466359, 6.54982041597820559567011241123, 7.23178649093466573525216516984, 7.82510917362072169206666260476, 9.016234936880634690360374570293, 9.597319049807815665446440370236

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