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Properties

Label 2-588-28.19-c1-0-5
Degree $2$
Conductor $588$
Sign $-0.661 - 0.749i$
Analytic cond. $4.69520$
Root an. cond. $2.16684$
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  + (−0.272 + 1.38i)2-s + (−0.5 − 0.866i)3-s + (−1.85 − 0.755i)4-s + (−2.12 − 1.22i)5-s + (1.33 − 0.458i)6-s + (1.55 − 2.36i)8-s + (−0.499 + 0.866i)9-s + (2.27 − 2.61i)10-s + (1.09 − 0.632i)11-s + (0.272 + 1.98i)12-s + 2.99i·13-s + 2.45i·15-s + (2.85 + 2.79i)16-s + (−1.58 + 0.916i)17-s + (−1.06 − 0.929i)18-s + (−2.07 + 3.60i)19-s + ⋯
L(s)  = 1  + (−0.192 + 0.981i)2-s + (−0.288 − 0.499i)3-s + (−0.925 − 0.377i)4-s + (−0.949 − 0.548i)5-s + (0.546 − 0.187i)6-s + (0.548 − 0.836i)8-s + (−0.166 + 0.288i)9-s + (0.720 − 0.826i)10-s + (0.330 − 0.190i)11-s + (0.0785 + 0.571i)12-s + 0.831i·13-s + 0.633i·15-s + (0.714 + 0.699i)16-s + (−0.385 + 0.222i)17-s + (−0.251 − 0.219i)18-s + (−0.477 + 0.826i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.661 - 0.749i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.661 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.221882 + 0.491977i\)
\(L(\frac12)\) \(\approx\) \(0.221882 + 0.491977i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.272 - 1.38i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (2.12 + 1.22i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.09 + 0.632i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.99iT - 13T^{2} \)
17 \( 1 + (1.58 - 0.916i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.07 - 3.60i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.83 - 3.36i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.42T + 29T^{2} \)
31 \( 1 + (-4.71 - 8.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.75 - 6.50i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.08iT - 41T^{2} \)
43 \( 1 - 6.27iT - 43T^{2} \)
47 \( 1 + (-3.67 + 6.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0358 - 0.0620i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.68 - 2.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.61 - 5.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.43 + 1.40i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.92iT - 71T^{2} \)
73 \( 1 + (7.01 - 4.05i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.54 - 0.891i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.33T + 83T^{2} \)
89 \( 1 + (7.42 + 4.28i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.10iT - 97T^{2} \)
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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

−11.12259263069439895470841703928, −9.955387663760086257341338994839, −8.817012022720835810550913600434, −8.393890238562032053044264432833, −7.33732433826514386955888251975, −6.72616059539004467142063347461, −5.64604691150239837719679025967, −4.63884515983790031468666309169, −3.71042282141582152717911349319, −1.35565825997104066407412140908, 0.37269891647278455313406012649, 2.51242558148729858285413205028, 3.60859987240446128047443573699, 4.38571792560063248575416743454, 5.47225616547493404100679059344, 6.95889646664199623205330320012, 7.85635527093255693318427702976, 8.885270882690259621949018041028, 9.603902714751109333791989020110, 10.73790085490045814635783216664

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