Properties

Label 2-588-196.167-c1-0-35
Degree $2$
Conductor $588$
Sign $-0.147 + 0.989i$
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  + (−1.20 − 0.732i)2-s + (−0.222 − 0.974i)3-s + (0.926 + 1.77i)4-s + (2.14 − 0.490i)5-s + (−0.445 + 1.34i)6-s + (1.67 − 2.05i)7-s + (0.178 − 2.82i)8-s + (−0.900 + 0.433i)9-s + (−2.95 − 0.980i)10-s + (1.39 − 2.88i)11-s + (1.52 − 1.29i)12-s + (−1.12 + 2.32i)13-s + (−3.52 + 1.25i)14-s + (−0.956 − 1.98i)15-s + (−2.28 + 3.28i)16-s + (1.59 + 1.27i)17-s + ⋯
L(s)  = 1  + (−0.855 − 0.518i)2-s + (−0.128 − 0.562i)3-s + (0.463 + 0.886i)4-s + (0.960 − 0.219i)5-s + (−0.181 + 0.548i)6-s + (0.631 − 0.775i)7-s + (0.0629 − 0.998i)8-s + (−0.300 + 0.144i)9-s + (−0.935 − 0.310i)10-s + (0.419 − 0.871i)11-s + (0.439 − 0.374i)12-s + (−0.311 + 0.645i)13-s + (−0.941 + 0.336i)14-s + (−0.246 − 0.512i)15-s + (−0.570 + 0.820i)16-s + (0.386 + 0.308i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.147 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.147 + 0.989i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.147 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.780228 - 0.904923i\)
\(L(\frac12)\) \(\approx\) \(0.780228 - 0.904923i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 + 0.732i)T \)
3 \( 1 + (0.222 + 0.974i)T \)
7 \( 1 + (-1.67 + 2.05i)T \)
good5 \( 1 + (-2.14 + 0.490i)T + (4.50 - 2.16i)T^{2} \)
11 \( 1 + (-1.39 + 2.88i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (1.12 - 2.32i)T + (-8.10 - 10.1i)T^{2} \)
17 \( 1 + (-1.59 - 1.27i)T + (3.78 + 16.5i)T^{2} \)
19 \( 1 - 2.54T + 19T^{2} \)
23 \( 1 + (-1.26 + 1.01i)T + (5.11 - 22.4i)T^{2} \)
29 \( 1 + (-4.92 + 6.17i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 - 4.58T + 31T^{2} \)
37 \( 1 + (5.04 - 6.32i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + (1.58 - 0.362i)T + (36.9 - 17.7i)T^{2} \)
43 \( 1 + (6.91 + 1.57i)T + (38.7 + 18.6i)T^{2} \)
47 \( 1 + (7.79 + 3.75i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-3.59 - 4.50i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (-2.42 + 10.6i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (5.04 + 4.02i)T + (13.5 + 59.4i)T^{2} \)
67 \( 1 + 2.06iT - 67T^{2} \)
71 \( 1 + (0.0808 - 0.0644i)T + (15.7 - 69.2i)T^{2} \)
73 \( 1 + (-5.32 - 11.0i)T + (-45.5 + 57.0i)T^{2} \)
79 \( 1 - 0.363iT - 79T^{2} \)
83 \( 1 + (2.97 - 1.43i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (3.25 + 6.76i)T + (-55.4 + 69.5i)T^{2} \)
97 \( 1 + 2.62iT - 97T^{2} \)
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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

−10.26245160143695410421388938273, −9.774765569895923186808178012861, −8.631715831862630849455210367934, −8.038301124263199703652878610351, −6.93639748004058719880652496141, −6.20506086904469799279977964087, −4.82625623713081652589005898044, −3.39990337149833024672201554706, −1.97563565698058792232284256123, −1.00967785298605851063817234080, 1.60309442666471708793764072970, 2.82855273885719982532533678914, 4.89026972239654710019986461885, 5.45172105018243736180862500505, 6.42685664187714462537010329731, 7.42488291041456728305888440332, 8.469518728072369893070270606032, 9.280489153347795907900175828084, 9.953068294292503644681206080313, 10.52574609703362336132506615563

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