Properties

Label 2-40e2-40.29-c1-0-19
Degree $2$
Conductor $1600$
Sign $0.748 + 0.663i$
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  − 0.732·3-s + 1.26i·7-s − 2.46·9-s − 3.46i·11-s + 3.46·13-s + 3.46i·17-s + 2i·19-s − 0.928i·21-s − 8.19i·23-s + 4·27-s − 9.46·31-s + 2.53i·33-s + 6·37-s − 2.53·39-s + 2.53·41-s + ⋯
L(s)  = 1  − 0.422·3-s + 0.479i·7-s − 0.821·9-s − 1.04i·11-s + 0.960·13-s + 0.840i·17-s + 0.458i·19-s − 0.202i·21-s − 1.70i·23-s + 0.769·27-s − 1.69·31-s + 0.441i·33-s + 0.986·37-s − 0.406·39-s + 0.396·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.748 + 0.663i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.748 + 0.663i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.245378060\)
\(L(\frac12)\) \(\approx\) \(1.245378060\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.732T + 3T^{2} \)
7 \( 1 - 1.26iT - 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 - 3.46iT - 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + 8.19iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 9.46T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 2.53T + 41T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 + 8.19iT - 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 + 12.9iT - 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 + 4.39T + 71T^{2} \)
73 \( 1 - 14.3iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 - 4.73T + 83T^{2} \)
89 \( 1 + 0.928T + 89T^{2} \)
97 \( 1 + 6.39iT - 97T^{2} \)
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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

−9.003036863942755256568299135340, −8.645430558527555678479659101735, −7.930103581680832938509215246999, −6.65948532230546944241151172917, −5.86448894286268259036193452799, −5.59332107084300178945845068308, −4.21221432115153884411414181234, −3.31909520095100338431866862732, −2.21168102323827038235937310569, −0.63154940203184196003722781810, 1.03133021652598914611604034506, 2.44557022853871952542399460665, 3.59918204684581914792743069194, 4.52205395001125350466114157894, 5.51193255205727252338744014877, 6.11056806771401984182229514169, 7.34847334894503915981212498640, 7.55687965073468735720905853312, 9.015847147436326252942076484211, 9.271699234839596304784165759783

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