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Properties

Label 2-592-37.36-c1-0-8
Degree $2$
Conductor $592$
Sign $0.657 - 0.753i$
Analytic cond. $4.72714$
Root an. cond. $2.17419$
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  + 2.79·3-s + 3.79i·5-s + 2·7-s + 4.79·9-s − 3.79·11-s − 0.791i·13-s + 10.5i·15-s + 1.58i·17-s − 7.58i·19-s + 5.58·21-s + 0.791i·23-s − 9.37·25-s + 4.99·27-s + 0.791i·29-s + 5.37i·31-s + ⋯
L(s)  = 1  + 1.61·3-s + 1.69i·5-s + 0.755·7-s + 1.59·9-s − 1.14·11-s − 0.219i·13-s + 2.73i·15-s + 0.383i·17-s − 1.73i·19-s + 1.21·21-s + 0.164i·23-s − 1.87·25-s + 0.962·27-s + 0.146i·29-s + 0.965i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.657 - 0.753i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ 0.657 - 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28823 + 1.03999i\)
\(L(\frac12)\) \(\approx\) \(2.28823 + 1.03999i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (-4 + 4.58i)T \)
good3 \( 1 - 2.79T + 3T^{2} \)
5 \( 1 - 3.79iT - 5T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + 3.79T + 11T^{2} \)
13 \( 1 + 0.791iT - 13T^{2} \)
17 \( 1 - 1.58iT - 17T^{2} \)
19 \( 1 + 7.58iT - 19T^{2} \)
23 \( 1 - 0.791iT - 23T^{2} \)
29 \( 1 - 0.791iT - 29T^{2} \)
31 \( 1 - 5.37iT - 31T^{2} \)
41 \( 1 - 5.20T + 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 + 1.58T + 47T^{2} \)
53 \( 1 - 7.58T + 53T^{2} \)
59 \( 1 + 7.58iT - 59T^{2} \)
61 \( 1 + 8.20iT - 61T^{2} \)
67 \( 1 - 7.37T + 67T^{2} \)
71 \( 1 + 9.16T + 71T^{2} \)
73 \( 1 - 9.37T + 73T^{2} \)
79 \( 1 - 12.7iT - 79T^{2} \)
83 \( 1 - 3.16T + 83T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 - 4.41iT - 97T^{2} \)
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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

−10.76437385168598542587209178245, −9.943976650875298572018903449641, −8.960448976495421937710713202422, −8.029310424406167477768817297598, −7.46241667496904362526364744931, −6.66980156396263579871873322055, −5.14552225760802228717635758772, −3.76836489061504928747261842618, −2.80354951918123124725201409542, −2.24338560013664233745101222465, 1.40329815882502554229838315588, 2.50443948228635396847976477443, 3.96385843605982555696851986362, 4.74884580534859068582072587296, 5.76814662378165667201976409888, 7.71067053100143453698458355835, 7.987870977431657266031803856395, 8.665314799904765514549787556416, 9.472718656708158670931174726558, 10.18945838802618489187938440507

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