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Properties

Label 2-588-28.3-c1-0-27
Degree $2$
Conductor $588$
Sign $-0.0257 + 0.999i$
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  + (−0.914 + 1.07i)2-s + (0.5 − 0.866i)3-s + (−0.326 − 1.97i)4-s + (−0.441 + 0.254i)5-s + (0.476 + 1.33i)6-s + (2.42 + 1.45i)8-s + (−0.499 − 0.866i)9-s + (0.129 − 0.709i)10-s + (−3.57 − 2.06i)11-s + (−1.87 − 0.704i)12-s − 3.97i·13-s + 0.509i·15-s + (−3.78 + 1.28i)16-s + (4.27 + 2.46i)17-s + (1.39 + 0.253i)18-s + (−2.52 − 4.37i)19-s + ⋯
L(s)  = 1  + (−0.646 + 0.762i)2-s + (0.288 − 0.499i)3-s + (−0.163 − 0.986i)4-s + (−0.197 + 0.114i)5-s + (0.194 + 0.543i)6-s + (0.857 + 0.513i)8-s + (−0.166 − 0.288i)9-s + (0.0407 − 0.224i)10-s + (−1.07 − 0.622i)11-s + (−0.540 − 0.203i)12-s − 1.10i·13-s + 0.131i·15-s + (−0.946 + 0.321i)16-s + (1.03 + 0.598i)17-s + (0.327 + 0.0596i)18-s + (−0.579 − 1.00i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0257 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0257 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.0257 + 0.999i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.0257 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.472369 - 0.484713i\)
\(L(\frac12)\) \(\approx\) \(0.472369 - 0.484713i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.914 - 1.07i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (0.441 - 0.254i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.57 + 2.06i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.97iT - 13T^{2} \)
17 \( 1 + (-4.27 - 2.46i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.52 + 4.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.52 - 1.45i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.82T + 29T^{2} \)
31 \( 1 + (-2.85 + 4.95i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.00 + 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 11.7iT - 41T^{2} \)
43 \( 1 + 9.84iT - 43T^{2} \)
47 \( 1 + (-3.93 - 6.81i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.619 - 1.07i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.32 - 4.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.470 - 0.271i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.08 - 3.51i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.06iT - 71T^{2} \)
73 \( 1 + (0.724 + 0.418i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.08 + 0.629i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 + (-4.06 + 2.34i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.80iT - 97T^{2} \)
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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

−10.50886130551061200264675915532, −9.384710872947019850590286229748, −8.503105059798626258983370586429, −7.72374667585366835916588648499, −7.27257542230376937409077908568, −5.83084708262727309732948150438, −5.44624073446287576316296206490, −3.69159859729129050164752204322, −2.22667740668582183777027431400, −0.43993697041367942537956637573, 1.81690033589766725553930027312, 3.01583331368188810385627086504, 4.14029923881183196439256126727, 5.01399893869557678345825602503, 6.61458829014296170949341274375, 7.88097937238946742877403686891, 8.227860041930563373195754130167, 9.472960313415504800468095481202, 9.973903327858933973267595219231, 10.68533634082991416136929876930

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