Properties

Label 2-588-196.187-c1-0-9
Degree $2$
Conductor $588$
Sign $0.960 + 0.279i$
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  + (0.0531 − 1.41i)2-s + (−0.988 − 0.149i)3-s + (−1.99 − 0.150i)4-s + (−0.462 + 0.181i)5-s + (−0.263 + 1.38i)6-s + (−2.59 − 0.512i)7-s + (−0.318 + 2.81i)8-s + (0.955 + 0.294i)9-s + (0.231 + 0.662i)10-s + (1.33 + 4.31i)11-s + (1.94 + 0.445i)12-s + (3.68 + 0.840i)13-s + (−0.862 + 3.64i)14-s + (0.483 − 0.110i)15-s + (3.95 + 0.599i)16-s + (1.13 − 1.66i)17-s + ⋯
L(s)  = 1  + (0.0375 − 0.999i)2-s + (−0.570 − 0.0860i)3-s + (−0.997 − 0.0751i)4-s + (−0.206 + 0.0811i)5-s + (−0.107 + 0.567i)6-s + (−0.981 − 0.193i)7-s + (−0.112 + 0.993i)8-s + (0.318 + 0.0982i)9-s + (0.0732 + 0.209i)10-s + (0.401 + 1.30i)11-s + (0.562 + 0.128i)12-s + (1.02 + 0.233i)13-s + (−0.230 + 0.973i)14-s + (0.124 − 0.0285i)15-s + (0.988 + 0.149i)16-s + (0.275 − 0.404i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.960 + 0.279i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.960 + 0.279i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.877644 - 0.125322i\)
\(L(\frac12)\) \(\approx\) \(0.877644 - 0.125322i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0531 + 1.41i)T \)
3 \( 1 + (0.988 + 0.149i)T \)
7 \( 1 + (2.59 + 0.512i)T \)
good5 \( 1 + (0.462 - 0.181i)T + (3.66 - 3.40i)T^{2} \)
11 \( 1 + (-1.33 - 4.31i)T + (-9.08 + 6.19i)T^{2} \)
13 \( 1 + (-3.68 - 0.840i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 + (-1.13 + 1.66i)T + (-6.21 - 15.8i)T^{2} \)
19 \( 1 + (-0.488 + 0.846i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.39 + 3.51i)T + (-8.40 + 21.4i)T^{2} \)
29 \( 1 + (-8.70 + 4.19i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (-4.48 - 7.76i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.392 - 5.23i)T + (-36.5 - 5.51i)T^{2} \)
41 \( 1 + (-2.78 - 2.22i)T + (9.12 + 39.9i)T^{2} \)
43 \( 1 + (2.87 - 2.29i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (-4.41 - 4.09i)T + (3.51 + 46.8i)T^{2} \)
53 \( 1 + (-0.569 - 7.59i)T + (-52.4 + 7.89i)T^{2} \)
59 \( 1 + (-4.24 + 10.8i)T + (-43.2 - 40.1i)T^{2} \)
61 \( 1 + (-9.45 - 0.708i)T + (60.3 + 9.09i)T^{2} \)
67 \( 1 + (11.2 - 6.47i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.51 - 9.38i)T + (-44.2 - 55.5i)T^{2} \)
73 \( 1 + (2.71 + 2.92i)T + (-5.45 + 72.7i)T^{2} \)
79 \( 1 + (-5.03 - 2.90i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.39 + 10.4i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (0.636 - 2.06i)T + (-73.5 - 50.1i)T^{2} \)
97 \( 1 - 4.38iT - 97T^{2} \)
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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

−10.54509901556640990394068262310, −10.02323490383656944877698160977, −9.251063153042024362400733744825, −8.167709539595940888925013445818, −6.91703243348073505619185191946, −6.10834099941419408117212537641, −4.76313487331103024183734144888, −3.96980762594920437935619051681, −2.76266056169288274302622054309, −1.17450447394118386156621427729, 0.66279658967472519310089217784, 3.38712133965144660712210102967, 4.14705231339678600601798744543, 5.77319530009836563408848629621, 5.95555734117630606289027774815, 6.92562067208279519503842803856, 8.147112811362850322504255302758, 8.759094557642245727840147510456, 9.784855509480876063495928834962, 10.55638916085938435801784319000

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