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Properties

Label 2-588-21.20-c3-0-34
Degree $2$
Conductor $588$
Sign $-0.571 - 0.820i$
Analytic cond. $34.6931$
Root an. cond. $5.89008$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.03 − 1.28i)3-s − 19.3·5-s + (23.7 + 12.9i)9-s − 41.9i·11-s − 77.1i·13-s + (97.5 + 24.8i)15-s − 5.96·17-s − 68.1i·19-s + 35.5i·23-s + 250.·25-s + (−102. − 95.4i)27-s − 228. i·29-s − 72.0i·31-s + (−53.7 + 211. i)33-s + 69.3·37-s + ⋯
L(s)  = 1  + (−0.969 − 0.246i)3-s − 1.73·5-s + (0.878 + 0.478i)9-s − 1.14i·11-s − 1.64i·13-s + (1.67 + 0.427i)15-s − 0.0851·17-s − 0.823i·19-s + 0.322i·23-s + 2.00·25-s + (−0.733 − 0.680i)27-s − 1.46i·29-s − 0.417i·31-s + (−0.283 + 1.11i)33-s + 0.308·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.571 - 0.820i$
Analytic conductor: \(34.6931\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ -0.571 - 0.820i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2142142943\)
\(L(\frac12)\) \(\approx\) \(0.2142142943\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.03 + 1.28i)T \)
7 \( 1 \)
good5 \( 1 + 19.3T + 125T^{2} \)
11 \( 1 + 41.9iT - 1.33e3T^{2} \)
13 \( 1 + 77.1iT - 2.19e3T^{2} \)
17 \( 1 + 5.96T + 4.91e3T^{2} \)
19 \( 1 + 68.1iT - 6.85e3T^{2} \)
23 \( 1 - 35.5iT - 1.21e4T^{2} \)
29 \( 1 + 228. iT - 2.43e4T^{2} \)
31 \( 1 + 72.0iT - 2.97e4T^{2} \)
37 \( 1 - 69.3T + 5.06e4T^{2} \)
41 \( 1 + 132.T + 6.89e4T^{2} \)
43 \( 1 + 366.T + 7.95e4T^{2} \)
47 \( 1 + 181.T + 1.03e5T^{2} \)
53 \( 1 + 17.6iT - 1.48e5T^{2} \)
59 \( 1 - 494.T + 2.05e5T^{2} \)
61 \( 1 - 575. iT - 2.26e5T^{2} \)
67 \( 1 + 404.T + 3.00e5T^{2} \)
71 \( 1 + 489. iT - 3.57e5T^{2} \)
73 \( 1 - 388. iT - 3.89e5T^{2} \)
79 \( 1 - 253.T + 4.93e5T^{2} \)
83 \( 1 - 356.T + 5.71e5T^{2} \)
89 \( 1 + 1.42e3T + 7.04e5T^{2} \)
97 \( 1 + 694. iT - 9.12e5T^{2} \)
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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

−9.994211904922761320733977532370, −8.462050842371205266875372181571, −7.923845455399238293913514421830, −7.11124541081034382708186468005, −6.01328956649969192153837070827, −5.07621451728668222722268913408, −4.01955592451829000900086176342, −3.00456969659347311028396468760, −0.76627010068147160927620732579, −0.11453443179594399573839156868, 1.56062336692726040960130452546, 3.59681620869548234142096524272, 4.36996449417082036924796199386, 5.03887137887832541291397941681, 6.69074696986221005520547494652, 7.04446458809750726323383624337, 8.113832504746327648101384297419, 9.161566480769155410173346030567, 10.15672568949937118726117703207, 11.04171825872909376862291324917

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