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Properties

Label 2-592-37.7-c1-0-17
Degree $2$
Conductor $592$
Sign $-0.785 - 0.618i$
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  + (−1.72 − 0.627i)3-s + (−0.326 − 1.85i)5-s + (−0.598 − 3.39i)7-s + (0.278 + 0.233i)9-s + (−1.40 + 2.43i)11-s + (−2.65 + 2.23i)13-s + (−0.598 + 3.39i)15-s + (−2.37 − 1.99i)17-s + (6.99 + 2.54i)19-s + (−1.09 + 6.22i)21-s + (−0.321 − 0.557i)23-s + (1.37 − 0.502i)25-s + (2.41 + 4.18i)27-s + (−1.08 + 1.87i)29-s − 9.90·31-s + ⋯
L(s)  = 1  + (−0.994 − 0.362i)3-s + (−0.145 − 0.827i)5-s + (−0.226 − 1.28i)7-s + (0.0927 + 0.0777i)9-s + (−0.423 + 0.733i)11-s + (−0.737 + 0.618i)13-s + (−0.154 + 0.876i)15-s + (−0.575 − 0.483i)17-s + (1.60 + 0.584i)19-s + (−0.239 + 1.35i)21-s + (−0.0670 − 0.116i)23-s + (0.275 − 0.100i)25-s + (0.465 + 0.805i)27-s + (−0.201 + 0.348i)29-s − 1.77·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 - 0.618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.785 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $-0.785 - 0.618i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ -0.785 - 0.618i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0555296 + 0.160197i\)
\(L(\frac12)\) \(\approx\) \(0.0555296 + 0.160197i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.56 - 5.51i)T \)
good3 \( 1 + (1.72 + 0.627i)T + (2.29 + 1.92i)T^{2} \)
5 \( 1 + (0.326 + 1.85i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.598 + 3.39i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (1.40 - 2.43i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.65 - 2.23i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (2.37 + 1.99i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (-6.99 - 2.54i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (0.321 + 0.557i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.08 - 1.87i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.90T + 31T^{2} \)
41 \( 1 + (8.13 - 6.82i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + 8.30T + 43T^{2} \)
47 \( 1 + (3.92 + 6.80i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.839 + 4.76i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (0.0961 - 0.545i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-5.37 + 4.50i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (0.0366 + 0.207i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-4.75 - 1.72i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 - 10.2T + 73T^{2} \)
79 \( 1 + (-0.330 - 1.87i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (5.21 + 4.37i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (1.68 - 9.54i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (8.52 + 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.11514866998905404543240877022, −9.520761784342611072974684915646, −8.282953893786050729659165508167, −7.10550273051923759466317542771, −6.84777924360013940804820006249, −5.26593027281354146984267029091, −4.83114701595258386785769184454, −3.50837208931461105337873385834, −1.49847696174310407619949094919, −0.10725204733640160672663848098, 2.48131667819566063336522937864, 3.43544504688807205731425433462, 5.27688952953003390725792096821, 5.45165098024130039250570549882, 6.56229362129491198361768016276, 7.56070408767171836447352683022, 8.680620850830896123856080285091, 9.594930974952323270156388984116, 10.59291702383045778031546005916, 11.14947815130202791271716000135

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