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Properties

Label 2-588-28.19-c1-0-11
Degree $2$
Conductor $588$
Sign $-0.510 - 0.859i$
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  + (0.772 + 1.18i)2-s + (−0.5 − 0.866i)3-s + (−0.807 + 1.82i)4-s + (2.68 + 1.55i)5-s + (0.639 − 1.26i)6-s + (−2.79 + 0.457i)8-s + (−0.499 + 0.866i)9-s + (0.237 + 4.38i)10-s + (−4.62 + 2.67i)11-s + (1.98 − 0.215i)12-s + 3.92i·13-s − 3.10i·15-s + (−2.69 − 2.95i)16-s + (4.92 − 2.84i)17-s + (−1.41 + 0.0764i)18-s + (−0.0854 + 0.147i)19-s + ⋯
L(s)  = 1  + (0.546 + 0.837i)2-s + (−0.288 − 0.499i)3-s + (−0.403 + 0.914i)4-s + (1.20 + 0.694i)5-s + (0.261 − 0.514i)6-s + (−0.986 + 0.161i)8-s + (−0.166 + 0.288i)9-s + (0.0750 + 1.38i)10-s + (−1.39 + 0.805i)11-s + (0.573 − 0.0623i)12-s + 1.08i·13-s − 0.801i·15-s + (−0.674 − 0.738i)16-s + (1.19 − 0.689i)17-s + (−0.332 + 0.0180i)18-s + (−0.0195 + 0.0339i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.510 - 0.859i$
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: \((2,\ 588,\ (\ :1/2),\ -0.510 - 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.888210 + 1.56065i\)
\(L(\frac12)\) \(\approx\) \(0.888210 + 1.56065i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.772 - 1.18i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-2.68 - 1.55i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.62 - 2.67i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.92iT - 13T^{2} \)
17 \( 1 + (-4.92 + 2.84i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0854 - 0.147i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.31 - 3.06i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.96T + 29T^{2} \)
31 \( 1 + (-0.765 - 1.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.46 - 7.74i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.84iT - 41T^{2} \)
43 \( 1 + 2.38iT - 43T^{2} \)
47 \( 1 + (1.14 - 1.98i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.599 - 1.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.49 + 9.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.70 - 3.29i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.35 + 4.82i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.04iT - 71T^{2} \)
73 \( 1 + (-8.89 + 5.13i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-12.4 - 7.19i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + (5.03 + 2.90i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.32iT - 97T^{2} \)
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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

−11.03161196121323683112361931116, −9.983861725793761954341662148828, −9.282060986510039152677298657995, −7.970241264291624537562826397617, −7.13856365265929028462213400071, −6.57781344243151550016338325766, −5.46540882761776537502435800130, −4.97030611008831109946894188163, −3.19715676128488998234971600711, −2.10407849040976882468366384348, 0.895485323617363725283358493288, 2.50337552419158034599376296129, 3.53385071596747732449438901421, 5.11974526702136441919566143558, 5.38395821649785399003164983924, 6.17858975377670164553531859667, 8.016568270776705102375100040373, 8.960136913261140267297798478827, 9.849892707353673217908982542573, 10.48716301787955042210366177038

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