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Properties

Label 2-588-7.6-c2-0-7
Degree $2$
Conductor $588$
Sign $0.987 + 0.156i$
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  + 1.73i·3-s − 0.929i·5-s − 2.99·9-s + 9.68·11-s − 15.9i·13-s + 1.60·15-s − 10.5i·17-s + 7.22i·19-s − 11.3·23-s + 24.1·25-s − 5.19i·27-s + 46.3·29-s + 0.483i·31-s + 16.7i·33-s + 2.48·37-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.185i·5-s − 0.333·9-s + 0.880·11-s − 1.22i·13-s + 0.107·15-s − 0.620i·17-s + 0.380i·19-s − 0.491·23-s + 0.965·25-s − 0.192i·27-s + 1.59·29-s + 0.0155i·31-s + 0.508i·33-s + 0.0670·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.987 + 0.156i$
Analytic conductor: \(16.0218\)
Root analytic conductor: \(4.00272\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1),\ 0.987 + 0.156i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.827460848\)
\(L(\frac12)\) \(\approx\) \(1.827460848\)
\(L(2)\) 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 \( 1 - 1.73iT \)
7 \( 1 \)
good5 \( 1 + 0.929iT - 25T^{2} \)
11 \( 1 - 9.68T + 121T^{2} \)
13 \( 1 + 15.9iT - 169T^{2} \)
17 \( 1 + 10.5iT - 289T^{2} \)
19 \( 1 - 7.22iT - 361T^{2} \)
23 \( 1 + 11.3T + 529T^{2} \)
29 \( 1 - 46.3T + 841T^{2} \)
31 \( 1 - 0.483iT - 961T^{2} \)
37 \( 1 - 2.48T + 1.36e3T^{2} \)
41 \( 1 + 55.8iT - 1.68e3T^{2} \)
43 \( 1 - 60.6T + 1.84e3T^{2} \)
47 \( 1 - 36.5iT - 2.20e3T^{2} \)
53 \( 1 + 28.5T + 2.80e3T^{2} \)
59 \( 1 + 94.0iT - 3.48e3T^{2} \)
61 \( 1 - 110. iT - 3.72e3T^{2} \)
67 \( 1 - 82.0T + 4.48e3T^{2} \)
71 \( 1 - 127.T + 5.04e3T^{2} \)
73 \( 1 - 46.2iT - 5.32e3T^{2} \)
79 \( 1 - 18.7T + 6.24e3T^{2} \)
83 \( 1 - 59.6iT - 6.88e3T^{2} \)
89 \( 1 + 71.1iT - 7.92e3T^{2} \)
97 \( 1 + 102. iT - 9.40e3T^{2} \)
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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

−10.41856968981285048256515489794, −9.640595991274611049955197558717, −8.774528938633902856136811125283, −7.968206922527325696051652369564, −6.81761939114577384711718276505, −5.78265010931383189385870532704, −4.84884351354011131782888780674, −3.80707125905884816500087823908, −2.69604568941333297642781411420, −0.855493466439144246883627151998, 1.16967114739148028839120010458, 2.43784333514702167578309107658, 3.82424170720474956032470904774, 4.87156275502027059765697560038, 6.39970905153446446215537327911, 6.63811467990137664679944661184, 7.84403929406885212517278390382, 8.763254145885257310811380619250, 9.504694149087621833121485337454, 10.61824238389770847526853672450

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