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Properties

Label 2-40e2-80.29-c1-0-14
Degree $2$
Conductor $1600$
Sign $0.755 + 0.655i$
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  + (−2.15 + 2.15i)3-s − 2.31·7-s − 6.31i·9-s + (−3.15 + 3.15i)11-s + (−4.31 + 4.31i)13-s + 1.31i·17-s + (−0.158 − 0.158i)19-s + (5 − 5i)21-s + 0.316·23-s + (7.15 + 7.15i)27-s + (−2 − 2i)29-s + 2.31·31-s − 13.6i·33-s + (0.683 + 0.683i)37-s − 18.6i·39-s + ⋯
L(s)  = 1  + (−1.24 + 1.24i)3-s − 0.875·7-s − 2.10i·9-s + (−0.952 + 0.952i)11-s + (−1.19 + 1.19i)13-s + 0.319i·17-s + (−0.0363 − 0.0363i)19-s + (1.09 − 1.09i)21-s + 0.0660·23-s + (1.37 + 1.37i)27-s + (−0.371 − 0.371i)29-s + 0.416·31-s − 2.37i·33-s + (0.112 + 0.112i)37-s − 2.98i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.755 + 0.655i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.755 + 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1425643733\)
\(L(\frac12)\) \(\approx\) \(0.1425643733\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.15 - 2.15i)T - 3iT^{2} \)
7 \( 1 + 2.31T + 7T^{2} \)
11 \( 1 + (3.15 - 3.15i)T - 11iT^{2} \)
13 \( 1 + (4.31 - 4.31i)T - 13iT^{2} \)
17 \( 1 - 1.31iT - 17T^{2} \)
19 \( 1 + (0.158 + 0.158i)T + 19iT^{2} \)
23 \( 1 - 0.316T + 23T^{2} \)
29 \( 1 + (2 + 2i)T + 29iT^{2} \)
31 \( 1 - 2.31T + 31T^{2} \)
37 \( 1 + (-0.683 - 0.683i)T + 37iT^{2} \)
41 \( 1 + 5iT - 41T^{2} \)
43 \( 1 + (7.63 + 7.63i)T + 43iT^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + (-3.31 - 3.31i)T + 53iT^{2} \)
59 \( 1 + (1.31 - 1.31i)T - 59iT^{2} \)
61 \( 1 + (-9.63 - 9.63i)T + 61iT^{2} \)
67 \( 1 + (-9.15 + 9.15i)T - 67iT^{2} \)
71 \( 1 + 8.63iT - 71T^{2} \)
73 \( 1 + 6.68T + 73T^{2} \)
79 \( 1 - 4.31T + 79T^{2} \)
83 \( 1 + (-7.15 + 7.15i)T - 83iT^{2} \)
89 \( 1 - 3.94iT - 89T^{2} \)
97 \( 1 + 6.63iT - 97T^{2} \)
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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

−9.690395694985567966652451958250, −8.897831630170076456882741381280, −7.48250433491511633635773815926, −6.74578221472860655166256346426, −5.95408983375833606487269765453, −5.01820908323059076375563169390, −4.53121800582566199119998475308, −3.59204921142588349302975206673, −2.27146499289248078884599076887, −0.093363555679981067424074398872, 0.76566600049742671456177411934, 2.36926723439860301322166052137, 3.25104307527819371630903866192, 5.05196913864700507464798102827, 5.43596447369959346599853739672, 6.32249683890767939218438704300, 6.96160185690865698015753500902, 7.77674752689003720997017539214, 8.350922380817250810424892825797, 9.798775821807177050906156483987

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