Properties

Label 2-588-196.87-c1-0-44
Degree $2$
Conductor $588$
Sign $0.741 - 0.670i$
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  + (1.26 + 0.637i)2-s + (0.988 − 0.149i)3-s + (1.18 + 1.60i)4-s + (1.46 + 0.575i)5-s + (1.34 + 0.442i)6-s + (1.59 − 2.11i)7-s + (0.473 + 2.78i)8-s + (0.955 − 0.294i)9-s + (1.48 + 1.65i)10-s + (−0.356 + 1.15i)11-s + (1.41 + 1.41i)12-s + (−4.55 + 1.04i)13-s + (3.35 − 1.65i)14-s + (1.53 + 0.350i)15-s + (−1.17 + 3.82i)16-s + (−3.40 − 4.98i)17-s + ⋯
L(s)  = 1  + (0.892 + 0.450i)2-s + (0.570 − 0.0860i)3-s + (0.593 + 0.804i)4-s + (0.655 + 0.257i)5-s + (0.548 + 0.180i)6-s + (0.601 − 0.799i)7-s + (0.167 + 0.985i)8-s + (0.318 − 0.0982i)9-s + (0.469 + 0.524i)10-s + (−0.107 + 0.348i)11-s + (0.408 + 0.408i)12-s + (−1.26 + 0.288i)13-s + (0.896 − 0.442i)14-s + (0.396 + 0.0904i)15-s + (−0.294 + 0.955i)16-s + (−0.824 − 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 - 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.741 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.741 - 0.670i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.741 - 0.670i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.02994 + 1.16652i\)
\(L(\frac12)\) \(\approx\) \(3.02994 + 1.16652i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 - 0.637i)T \)
3 \( 1 + (-0.988 + 0.149i)T \)
7 \( 1 + (-1.59 + 2.11i)T \)
good5 \( 1 + (-1.46 - 0.575i)T + (3.66 + 3.40i)T^{2} \)
11 \( 1 + (0.356 - 1.15i)T + (-9.08 - 6.19i)T^{2} \)
13 \( 1 + (4.55 - 1.04i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (3.40 + 4.98i)T + (-6.21 + 15.8i)T^{2} \)
19 \( 1 + (-1.30 - 2.26i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.84 + 5.63i)T + (-8.40 - 21.4i)T^{2} \)
29 \( 1 + (1.17 + 0.563i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (1.77 - 3.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.292 - 3.90i)T + (-36.5 + 5.51i)T^{2} \)
41 \( 1 + (3.78 - 3.01i)T + (9.12 - 39.9i)T^{2} \)
43 \( 1 + (-3.36 - 2.68i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (0.449 - 0.417i)T + (3.51 - 46.8i)T^{2} \)
53 \( 1 + (0.651 - 8.69i)T + (-52.4 - 7.89i)T^{2} \)
59 \( 1 + (0.100 + 0.256i)T + (-43.2 + 40.1i)T^{2} \)
61 \( 1 + (7.65 - 0.573i)T + (60.3 - 9.09i)T^{2} \)
67 \( 1 + (-9.08 - 5.24i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.56 + 11.5i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-7.52 + 8.10i)T + (-5.45 - 72.7i)T^{2} \)
79 \( 1 + (-6.37 + 3.68i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.54 + 6.75i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (3.13 + 10.1i)T + (-73.5 + 50.1i)T^{2} \)
97 \( 1 - 8.77iT - 97T^{2} \)
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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

−10.84424952620552041575118741078, −9.953578291569187221626874075518, −8.959605732540462118389572159428, −7.78510196519698972397547532518, −7.18833863685630072619064105434, −6.40814051975885106321393217684, −4.96157272195518599796787154874, −4.46579590180324557318765275941, −2.98343441486222983869097688841, −2.03843263809148630872803243756, 1.77690337980246975440406581093, 2.58378473286767668543846089476, 3.86103237356421612272080827316, 5.13561870430585816834267354762, 5.57740664514863019927183851214, 6.86465363834819875892421879659, 7.942273259336602932155925970432, 9.126648857461954227737505286956, 9.654278267563711459153616048160, 10.76515838843366847732698395485

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