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Properties

Label 2-588-4.3-c2-0-14
Degree $2$
Conductor $588$
Sign $-0.884 + 0.467i$
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  + (1.71 + 1.03i)2-s + 1.73i·3-s + (1.86 + 3.53i)4-s − 2.98·5-s + (−1.78 + 2.96i)6-s + (−0.449 + 7.98i)8-s − 2.99·9-s + (−5.11 − 3.07i)10-s − 7.51i·11-s + (−6.12 + 3.23i)12-s − 23.3·13-s − 5.16i·15-s + (−9.01 + 13.2i)16-s − 10.3·17-s + (−5.13 − 3.09i)18-s + 15.5i·19-s + ⋯
L(s)  = 1  + (0.856 + 0.516i)2-s + 0.577i·3-s + (0.467 + 0.884i)4-s − 0.596·5-s + (−0.297 + 0.494i)6-s + (−0.0562 + 0.998i)8-s − 0.333·9-s + (−0.511 − 0.307i)10-s − 0.682i·11-s + (−0.510 + 0.269i)12-s − 1.79·13-s − 0.344i·15-s + (−0.563 + 0.826i)16-s − 0.606·17-s + (−0.285 − 0.172i)18-s + 0.817i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.467i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.884 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.884 + 0.467i$
Analytic conductor: \(16.0218\)
Root analytic conductor: \(4.00272\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1),\ -0.884 + 0.467i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.137367888\)
\(L(\frac12)\) \(\approx\) \(1.137367888\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.71 - 1.03i)T \)
3 \( 1 - 1.73iT \)
7 \( 1 \)
good5 \( 1 + 2.98T + 25T^{2} \)
11 \( 1 + 7.51iT - 121T^{2} \)
13 \( 1 + 23.3T + 169T^{2} \)
17 \( 1 + 10.3T + 289T^{2} \)
19 \( 1 - 15.5iT - 361T^{2} \)
23 \( 1 - 27.5iT - 529T^{2} \)
29 \( 1 - 19.6T + 841T^{2} \)
31 \( 1 + 28.5iT - 961T^{2} \)
37 \( 1 - 53.4T + 1.36e3T^{2} \)
41 \( 1 + 31.1T + 1.68e3T^{2} \)
43 \( 1 + 15.2iT - 1.84e3T^{2} \)
47 \( 1 - 3.62iT - 2.20e3T^{2} \)
53 \( 1 + 68.8T + 2.80e3T^{2} \)
59 \( 1 - 113. iT - 3.48e3T^{2} \)
61 \( 1 + 3.30T + 3.72e3T^{2} \)
67 \( 1 + 92.9iT - 4.48e3T^{2} \)
71 \( 1 - 72.5iT - 5.04e3T^{2} \)
73 \( 1 - 69.3T + 5.32e3T^{2} \)
79 \( 1 - 156. iT - 6.24e3T^{2} \)
83 \( 1 + 85.1iT - 6.88e3T^{2} \)
89 \( 1 - 17.2T + 7.92e3T^{2} \)
97 \( 1 - 110.T + 9.40e3T^{2} \)
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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

−11.32508157006589739638313197450, −10.14005486565533871436064688041, −9.205346217497106738265140495847, −8.014603377150611924944398329298, −7.52652139068182740774477972231, −6.31523730617625812489159497959, −5.36257285460928577752427256240, −4.46439565393497628162325404289, −3.60264816095180976151511109379, −2.46979582103283477806141597106, 0.29340648022589124133679965664, 2.06288392226899603962644168322, 2.94450714214419289456536975218, 4.46445644518080731639306070016, 4.96148385706571094761891830029, 6.44055197846084875816853518584, 7.07314806826566732703140677928, 7.989328017317160254499240453213, 9.326296319621290856082906758341, 10.13996108593497967375069166913

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