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Properties

Label 2-588-21.2-c2-0-25
Degree $2$
Conductor $588$
Sign $-0.946 - 0.323i$
Analytic cond. $16.0218$
Root an. cond. $4.00272$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.68 − 1.34i)3-s + (−3.34 − 1.93i)5-s + (5.38 − 7.20i)9-s + (−12.2 + 7.05i)11-s − 20.2·13-s + (−11.5 − 0.685i)15-s + (−18.9 + 10.9i)17-s + (3.46 − 6.00i)19-s + (6.69 + 3.86i)23-s + (−5.02 − 8.69i)25-s + (4.76 − 26.5i)27-s + 37.3i·29-s + (7.64 + 13.2i)31-s + (−23.2 + 35.3i)33-s + (6.06 − 10.5i)37-s + ⋯
L(s)  = 1  + (0.894 − 0.447i)3-s + (−0.669 − 0.386i)5-s + (0.598 − 0.801i)9-s + (−1.11 + 0.640i)11-s − 1.55·13-s + (−0.772 − 0.0456i)15-s + (−1.11 + 0.642i)17-s + (0.182 − 0.316i)19-s + (0.291 + 0.168i)23-s + (−0.200 − 0.347i)25-s + (0.176 − 0.984i)27-s + 1.28i·29-s + (0.246 + 0.427i)31-s + (−0.705 + 1.07i)33-s + (0.163 − 0.283i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 - 0.323i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.946 - 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.946 - 0.323i$
Analytic conductor: \(16.0218\)
Root analytic conductor: \(4.00272\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1),\ -0.946 - 0.323i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1986921184\)
\(L(\frac12)\) \(\approx\) \(0.1986921184\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-2.68 + 1.34i)T \)
7 \( 1 \)
good5 \( 1 + (3.34 + 1.93i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (12.2 - 7.05i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 20.2T + 169T^{2} \)
17 \( 1 + (18.9 - 10.9i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-3.46 + 6.00i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-6.69 - 3.86i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 37.3iT - 841T^{2} \)
31 \( 1 + (-7.64 - 13.2i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-6.06 + 10.5i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 42.3iT - 1.68e3T^{2} \)
43 \( 1 - 16.1T + 1.84e3T^{2} \)
47 \( 1 + (62.2 + 35.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (87.8 - 50.7i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (22.2 - 12.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-39.0 + 67.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-61.5 - 106. i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 34.5iT - 5.04e3T^{2} \)
73 \( 1 + (50.2 + 87.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-21.2 + 36.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 82.1iT - 6.88e3T^{2} \)
89 \( 1 + (-36.6 - 21.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 48.3T + 9.40e3T^{2} \)
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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

−9.876606304136565473440889544774, −9.006433746315537127634571839879, −8.156202251138080096438927097501, −7.45045754510834571748192717486, −6.75353843670081927406622854923, −5.10436166371329437386538331698, −4.30708794191141907600020102246, −2.96348799859861117011862415796, −1.96485812943584707108312952070, −0.06057080629647432757729131527, 2.38386106660481815365713878108, 3.10376687300684622642069907284, 4.37097472338566566101195355177, 5.16017607041899197874806394978, 6.70001588574628375330280178783, 7.80461949509953185719738203537, 8.035330165096605850313147417916, 9.377729877797423299473798818534, 9.919947460620834833137274392304, 10.96682422255256465056686356474

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