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Properties

Label 2-588-84.11-c1-0-34
Degree $2$
Conductor $588$
Sign $0.999 - 0.0309i$
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  + (1.38 − 0.283i)2-s + (−1.62 + 0.593i)3-s + (1.83 − 0.786i)4-s + (−0.695 + 0.401i)5-s + (−2.08 + 1.28i)6-s + (2.32 − 1.61i)8-s + (2.29 − 1.93i)9-s + (−0.849 + 0.753i)10-s + (−1.17 + 2.03i)11-s + (−2.52 + 2.37i)12-s + 5.26·13-s + (0.893 − 1.06i)15-s + (2.76 − 2.89i)16-s + (1.02 + 0.592i)17-s + (2.63 − 3.32i)18-s + (6.16 − 3.56i)19-s + ⋯
L(s)  = 1  + (0.979 − 0.200i)2-s + (−0.939 + 0.342i)3-s + (0.919 − 0.393i)4-s + (−0.311 + 0.179i)5-s + (−0.851 + 0.524i)6-s + (0.821 − 0.569i)8-s + (0.765 − 0.643i)9-s + (−0.268 + 0.238i)10-s + (−0.353 + 0.612i)11-s + (−0.729 + 0.684i)12-s + 1.46·13-s + (0.230 − 0.275i)15-s + (0.690 − 0.723i)16-s + (0.248 + 0.143i)17-s + (0.620 − 0.784i)18-s + (1.41 − 0.817i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.999 - 0.0309i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.999 - 0.0309i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09521 + 0.0323873i\)
\(L(\frac12)\) \(\approx\) \(2.09521 + 0.0323873i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.283i)T \)
3 \( 1 + (1.62 - 0.593i)T \)
7 \( 1 \)
good5 \( 1 + (0.695 - 0.401i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.17 - 2.03i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.26T + 13T^{2} \)
17 \( 1 + (-1.02 - 0.592i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.16 + 3.56i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.94 - 6.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.23iT - 29T^{2} \)
31 \( 1 + (4.24 + 2.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.523 + 0.907i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.16iT - 41T^{2} \)
43 \( 1 + 7.94iT - 43T^{2} \)
47 \( 1 + (3.04 + 5.28i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.55 - 4.36i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.331 - 0.573i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.479 + 0.829i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.29 + 4.21i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.67T + 71T^{2} \)
73 \( 1 + (-0.707 + 1.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6 - 3.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.18T + 83T^{2} \)
89 \( 1 + (14.1 - 8.18i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.37T + 97T^{2} \)
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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

−11.14054885268824637956659549511, −10.10410789296752762396908236913, −9.278652320170459833394762966913, −7.58535838269688565195949330504, −6.96167385802535107272516447737, −5.78128338040339133953384337192, −5.25654580030020931319808776467, −4.07394346335679140716558758331, −3.26174655687667602523320074556, −1.36516788705904866837280355608, 1.28708719937587141416983760259, 3.10711947712216535593038672953, 4.21519403074273773851041973773, 5.31692422133669921109841087159, 5.94994135722780070849600411406, 6.85141854097617048128181119716, 7.78013101645667853995430321586, 8.639895196982462820539710833860, 10.30152546002148079398223622449, 10.96000476951616962294749175254

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