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Properties

Label 2-588-84.11-c1-0-59
Degree $2$
Conductor $588$
Sign $-0.856 + 0.515i$
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  + (0.938 − 1.05i)2-s + (−1.62 + 0.593i)3-s + (−0.238 − 1.98i)4-s + (0.695 − 0.401i)5-s + (−0.899 + 2.27i)6-s + (−2.32 − 1.61i)8-s + (2.29 − 1.93i)9-s + (0.228 − 1.11i)10-s + (1.17 − 2.03i)11-s + (1.56 + 3.09i)12-s − 5.26·13-s + (−0.893 + 1.06i)15-s + (−3.88 + 0.945i)16-s + (−1.02 − 0.592i)17-s + (0.112 − 4.24i)18-s + (6.16 − 3.56i)19-s + ⋯
L(s)  = 1  + (0.663 − 0.747i)2-s + (−0.939 + 0.342i)3-s + (−0.119 − 0.992i)4-s + (0.311 − 0.179i)5-s + (−0.367 + 0.930i)6-s + (−0.821 − 0.569i)8-s + (0.765 − 0.643i)9-s + (0.0721 − 0.351i)10-s + (0.353 − 0.612i)11-s + (0.451 + 0.892i)12-s − 1.46·13-s + (−0.230 + 0.275i)15-s + (−0.971 + 0.236i)16-s + (−0.248 − 0.143i)17-s + (0.0265 − 0.999i)18-s + (1.41 − 0.817i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.856 + 0.515i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.856 + 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.315518 - 1.13674i\)
\(L(\frac12)\) \(\approx\) \(0.315518 - 1.13674i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.938 + 1.05i)T \)
3 \( 1 + (1.62 - 0.593i)T \)
7 \( 1 \)
good5 \( 1 + (-0.695 + 0.401i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.17 + 2.03i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.26T + 13T^{2} \)
17 \( 1 + (1.02 + 0.592i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.16 + 3.56i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.94 + 6.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.23iT - 29T^{2} \)
31 \( 1 + (4.24 + 2.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.523 + 0.907i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.16iT - 41T^{2} \)
43 \( 1 - 7.94iT - 43T^{2} \)
47 \( 1 + (3.04 + 5.28i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.55 - 4.36i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.331 - 0.573i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.479 - 0.829i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.29 - 4.21i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.67T + 71T^{2} \)
73 \( 1 + (0.707 - 1.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6 + 3.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.18T + 83T^{2} \)
89 \( 1 + (-14.1 + 8.18i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.37T + 97T^{2} \)
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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

−10.43719828797916197931282040043, −9.712181321764165736480902511830, −9.100954839454996679780862978315, −7.37461556837298557795055613436, −6.36087087425944556089511696019, −5.45214032706725400913844922682, −4.79004849131324192935889591370, −3.72518705219118496129634044225, −2.30655215952180776917424543446, −0.58439509017899921756640128563, 2.02072707194465435770560674895, 3.67270690942346860565655987686, 4.92356496780301527648466352848, 5.51042392786855557551774411337, 6.51442035475577567434245853056, 7.31380251944162371296330633488, 7.88754811530155706925370887767, 9.466094835184655096660168593443, 10.08513319206028485094819582224, 11.40664873280439147620004841337

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