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Properties

Label 2-588-28.19-c1-0-22
Degree $2$
Conductor $588$
Sign $-0.0736 + 0.997i$
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  + (−0.381 − 1.36i)2-s + (0.5 + 0.866i)3-s + (−1.70 + 1.03i)4-s + (−0.977 − 0.564i)5-s + (0.988 − 1.01i)6-s + (2.06 + 1.93i)8-s + (−0.499 + 0.866i)9-s + (−0.396 + 1.54i)10-s + (−1.16 + 0.675i)11-s + (−1.75 − 0.961i)12-s − 5.58i·13-s − 1.12i·15-s + (1.84 − 3.54i)16-s + (3.44 − 1.98i)17-s + (1.36 + 0.350i)18-s + (3.07 − 5.32i)19-s + ⋯
L(s)  = 1  + (−0.269 − 0.963i)2-s + (0.288 + 0.499i)3-s + (−0.854 + 0.519i)4-s + (−0.437 − 0.252i)5-s + (0.403 − 0.412i)6-s + (0.730 + 0.683i)8-s + (−0.166 + 0.288i)9-s + (−0.125 + 0.489i)10-s + (−0.352 + 0.203i)11-s + (−0.506 − 0.277i)12-s − 1.54i·13-s − 0.291i·15-s + (0.461 − 0.887i)16-s + (0.834 − 0.481i)17-s + (0.322 + 0.0827i)18-s + (0.704 − 1.22i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0736 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0736 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.0736 + 0.997i$
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: \((2,\ 588,\ (\ :1/2),\ -0.0736 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.758169 - 0.816189i\)
\(L(\frac12)\) \(\approx\) \(0.758169 - 0.816189i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.381 + 1.36i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (0.977 + 0.564i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.16 - 0.675i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.58iT - 13T^{2} \)
17 \( 1 + (-3.44 + 1.98i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.07 + 5.32i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.72 - 3.30i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.55T + 29T^{2} \)
31 \( 1 + (1.46 + 2.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.16 - 5.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.149iT - 41T^{2} \)
43 \( 1 + 10.6iT - 43T^{2} \)
47 \( 1 + (1.79 - 3.11i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.366 + 0.635i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.20 + 12.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.89 - 4.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.202 + 0.117i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.74iT - 71T^{2} \)
73 \( 1 + (1.52 - 0.882i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.28 + 4.20i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.12T + 83T^{2} \)
89 \( 1 + (-6.86 - 3.96i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.23iT - 97T^{2} \)
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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

−10.33949272772952048159291616563, −9.821482506405376506283919164451, −8.819682285637669777174764454938, −8.079835171374811283005794180187, −7.28609994321344440749806795853, −5.35415957716752560211290735129, −4.75991225558548797626199852673, −3.40681173830893179018670058082, −2.74493685874087774220692600072, −0.76436286796510275325767152529, 1.40753583998653360180433696719, 3.31071337964351690880262170008, 4.47681050567990254448306897944, 5.67325624263536833952402087361, 6.63580754418826573646833856715, 7.36100038272766351808573848711, 8.143864438543925054175456517941, 8.924749275615030941045018950463, 9.820870263769081876426238557602, 10.77900503382363742375443712747

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