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Properties

Label 2-592-37.10-c1-0-6
Degree $2$
Conductor $592$
Sign $-0.351 - 0.936i$
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  + (1.42 + 2.47i)3-s + (0.5 + 0.866i)5-s + (2.07 + 3.59i)7-s + (−2.57 + 4.46i)9-s + 1.29·11-s + (−1.77 − 3.07i)13-s + (−1.42 + 2.47i)15-s + (1.79 − 3.11i)17-s + (−2.42 − 4.20i)19-s + (−5.93 + 10.2i)21-s − 5.01·23-s + (2 − 3.46i)25-s − 6.15·27-s + 8.86·29-s − 1.29·31-s + ⋯
L(s)  = 1  + (0.824 + 1.42i)3-s + (0.223 + 0.387i)5-s + (0.785 + 1.36i)7-s + (−0.859 + 1.48i)9-s + 0.391·11-s + (−0.493 − 0.854i)13-s + (−0.368 + 0.638i)15-s + (0.436 − 0.756i)17-s + (−0.556 − 0.964i)19-s + (−1.29 + 2.24i)21-s − 1.04·23-s + (0.400 − 0.692i)25-s − 1.18·27-s + 1.64·29-s − 0.233·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $-0.351 - 0.936i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ -0.351 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19235 + 1.72060i\)
\(L(\frac12)\) \(\approx\) \(1.19235 + 1.72060i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.27 - 5.94i)T \)
good3 \( 1 + (-1.42 - 2.47i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.07 - 3.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 1.29T + 11T^{2} \)
13 \( 1 + (1.77 + 3.07i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.79 + 3.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.42 + 4.20i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.01T + 23T^{2} \)
29 \( 1 - 8.86T + 29T^{2} \)
31 \( 1 + 1.29T + 31T^{2} \)
41 \( 1 + (5.35 + 9.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 + 8.72T + 47T^{2} \)
53 \( 1 + (5.07 - 8.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.63 - 8.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.35 + 2.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.98 - 6.90i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.63 + 9.75i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.28T + 73T^{2} \)
79 \( 1 + (-1.28 - 2.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.98 + 8.63i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.21 + 3.83i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.97T + 97T^{2} \)
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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

−10.64318649045893529469991551252, −10.07713235562716485526000430116, −9.133865976650755534796313967330, −8.610629624651229127269198893667, −7.75762728759862591053324919541, −6.23384823992869851387126638886, −5.12469673524202706910202323513, −4.50572119382934479572862400354, −3.03603364640843091421160148765, −2.41405817902743444667782395391, 1.22648081865298083817727439610, 1.96239056929688746112637619044, 3.61563060868960577655738816220, 4.64485844093311378852595170605, 6.23401624028005900617997258215, 6.95046381897456845764862325495, 7.933432204206587579150097859101, 8.228838297027302533499265561570, 9.407615633945734516296008167737, 10.36989726183418301739795401546

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