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Properties

Label 2-588-196.87-c1-0-22
Degree $2$
Conductor $588$
Sign $0.0419 - 0.999i$
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  + (−0.109 + 1.40i)2-s + (0.988 − 0.149i)3-s + (−1.97 − 0.309i)4-s + (0.577 + 0.226i)5-s + (0.101 + 1.41i)6-s + (2.12 + 1.57i)7-s + (0.654 − 2.75i)8-s + (0.955 − 0.294i)9-s + (−0.383 + 0.789i)10-s + (0.418 − 1.35i)11-s + (−1.99 − 0.0120i)12-s + (−0.804 + 0.183i)13-s + (−2.45 + 2.82i)14-s + (0.605 + 0.138i)15-s + (3.80 + 1.22i)16-s + (3.97 + 5.83i)17-s + ⋯
L(s)  = 1  + (−0.0777 + 0.996i)2-s + (0.570 − 0.0860i)3-s + (−0.987 − 0.154i)4-s + (0.258 + 0.101i)5-s + (0.0414 + 0.575i)6-s + (0.803 + 0.595i)7-s + (0.231 − 0.972i)8-s + (0.318 − 0.0982i)9-s + (−0.121 + 0.249i)10-s + (0.126 − 0.409i)11-s + (−0.577 − 0.00346i)12-s + (−0.223 + 0.0509i)13-s + (−0.656 + 0.754i)14-s + (0.156 + 0.0356i)15-s + (0.951 + 0.306i)16-s + (0.964 + 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0419 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0419 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.0419 - 0.999i$
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: \((2,\ 588,\ (\ :1/2),\ 0.0419 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28062 + 1.22803i\)
\(L(\frac12)\) \(\approx\) \(1.28062 + 1.22803i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.109 - 1.40i)T \)
3 \( 1 + (-0.988 + 0.149i)T \)
7 \( 1 + (-2.12 - 1.57i)T \)
good5 \( 1 + (-0.577 - 0.226i)T + (3.66 + 3.40i)T^{2} \)
11 \( 1 + (-0.418 + 1.35i)T + (-9.08 - 6.19i)T^{2} \)
13 \( 1 + (0.804 - 0.183i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (-3.97 - 5.83i)T + (-6.21 + 15.8i)T^{2} \)
19 \( 1 + (-0.990 - 1.71i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.55 + 3.74i)T + (-8.40 - 21.4i)T^{2} \)
29 \( 1 + (-2.01 - 0.970i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (2.30 - 3.98i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.849 - 11.3i)T + (-36.5 + 5.51i)T^{2} \)
41 \( 1 + (0.142 - 0.113i)T + (9.12 - 39.9i)T^{2} \)
43 \( 1 + (7.66 + 6.10i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (-9.80 + 9.09i)T + (3.51 - 46.8i)T^{2} \)
53 \( 1 + (0.412 - 5.50i)T + (-52.4 - 7.89i)T^{2} \)
59 \( 1 + (2.34 + 5.97i)T + (-43.2 + 40.1i)T^{2} \)
61 \( 1 + (9.20 - 0.689i)T + (60.3 - 9.09i)T^{2} \)
67 \( 1 + (0.491 + 0.283i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.11 + 6.46i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (2.32 - 2.50i)T + (-5.45 - 72.7i)T^{2} \)
79 \( 1 + (-4.26 + 2.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.542 - 2.37i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (0.190 + 0.616i)T + (-73.5 + 50.1i)T^{2} \)
97 \( 1 + 17.0iT - 97T^{2} \)
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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

−10.55353982772878202414428487401, −9.911550550119970576209495847828, −8.671924287320757803048858154478, −8.408373026121302264451432418488, −7.48025928331598216858924899927, −6.37103090952043314484891900165, −5.56383449574513305897788206227, −4.52850050589571348829512520721, −3.29294055887462204604281744478, −1.58953267908746872883569558776, 1.17956212449131017085870528439, 2.44192552963429455613591915119, 3.60763389021913529213065398290, 4.65578108650694555258127516898, 5.47156897051243759804711019919, 7.41727175532377305302738548801, 7.77512884966197100941671316690, 9.176081034702775456515252432136, 9.508792056953418958318315252812, 10.46862534823938511831632071931

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