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Properties

Label 2-592-37.11-c1-0-13
Degree $2$
Conductor $592$
Sign $0.443 + 0.896i$
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  + (1.36 − 2.36i)3-s + (1.5 + 0.866i)5-s + (−1 + 1.73i)7-s + (−2.23 − 3.86i)9-s + 4.73·11-s + (−3 − 1.73i)13-s + (4.09 − 2.36i)15-s + (6.69 − 3.86i)17-s + (−1.09 − 0.633i)19-s + (2.73 + 4.73i)21-s − 4.73i·23-s + (−1 − 1.73i)25-s − 4.00·27-s + 8.66i·29-s + 1.26i·31-s + ⋯
L(s)  = 1  + (0.788 − 1.36i)3-s + (0.670 + 0.387i)5-s + (−0.377 + 0.654i)7-s + (−0.744 − 1.28i)9-s + 1.42·11-s + (−0.832 − 0.480i)13-s + (1.05 − 0.610i)15-s + (1.62 − 0.937i)17-s + (−0.251 − 0.145i)19-s + (0.596 + 1.03i)21-s − 0.986i·23-s + (−0.200 − 0.346i)25-s − 0.769·27-s + 1.60i·29-s + 0.227i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.443 + 0.896i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ 0.443 + 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80159 - 1.11879i\)
\(L(\frac12)\) \(\approx\) \(1.80159 - 1.11879i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.69 + 2.13i)T \)
good3 \( 1 + (-1.36 + 2.36i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 4.73T + 11T^{2} \)
13 \( 1 + (3 + 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-6.69 + 3.86i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.09 + 0.633i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.73iT - 23T^{2} \)
29 \( 1 - 8.66iT - 29T^{2} \)
31 \( 1 - 1.26iT - 31T^{2} \)
41 \( 1 + (4.96 - 8.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 0.928iT - 43T^{2} \)
47 \( 1 + 4.73T + 47T^{2} \)
53 \( 1 + (1.26 + 2.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.19 - 1.26i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 0.866i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.09 - 8.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.73 + 3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + (-11.4 - 6.63i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.83 + 4.90i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.89 - 3.40i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.73iT - 97T^{2} \)
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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

−10.32661883506719194998945843576, −9.473944582428718355912184372610, −8.768309892268009058682854905807, −7.80170950817519754081108825748, −6.86524692381338767908854037187, −6.35849650452927331400811528648, −5.18494800891820305635030744786, −3.28875383947870717323617446545, −2.52637846742919298994091458937, −1.31319466683809854682714035615, 1.75892190641303370850394615058, 3.48851058270205280070355394743, 3.97039160964177468677306455853, 5.11565785556131063390584949370, 6.16453483572719737902115581695, 7.41458148162080071685790421824, 8.494827002453309158329776665719, 9.423338255012495276513788957363, 9.785971265707106308953896654304, 10.37393098734813912151371848150

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