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Properties

Label 2-588-28.27-c1-0-22
Degree $2$
Conductor $588$
Sign $0.891 + 0.453i$
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  + (−1.40 + 0.171i)2-s + 3-s + (1.94 − 0.481i)4-s + 0.963i·5-s + (−1.40 + 0.171i)6-s + (−2.64 + 1.00i)8-s + 9-s + (−0.165 − 1.35i)10-s − 5.48i·11-s + (1.94 − 0.481i)12-s − 3.75i·13-s + 0.963i·15-s + (3.53 − 1.87i)16-s + 0.686i·17-s + (−1.40 + 0.171i)18-s + 4.88·19-s + ⋯
L(s)  = 1  + (−0.992 + 0.121i)2-s + 0.577·3-s + (0.970 − 0.240i)4-s + 0.430i·5-s + (−0.573 + 0.0700i)6-s + (−0.934 + 0.356i)8-s + 0.333·9-s + (−0.0523 − 0.427i)10-s − 1.65i·11-s + (0.560 − 0.139i)12-s − 1.04i·13-s + 0.248i·15-s + (0.883 − 0.467i)16-s + 0.166i·17-s + (−0.330 + 0.0404i)18-s + 1.12·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.891 + 0.453i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.891 + 0.453i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13782 - 0.272665i\)
\(L(\frac12)\) \(\approx\) \(1.13782 - 0.272665i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.171i)T \)
3 \( 1 - T \)
7 \( 1 \)
good5 \( 1 - 0.963iT - 5T^{2} \)
11 \( 1 + 5.48iT - 11T^{2} \)
13 \( 1 + 3.75iT - 13T^{2} \)
17 \( 1 - 0.686iT - 17T^{2} \)
19 \( 1 - 4.88T + 19T^{2} \)
23 \( 1 + 1.24iT - 23T^{2} \)
29 \( 1 + 2.48T + 29T^{2} \)
31 \( 1 - 4.82T + 31T^{2} \)
37 \( 1 + 2.73T + 37T^{2} \)
41 \( 1 - 9.42iT - 41T^{2} \)
43 \( 1 + 5.97iT - 43T^{2} \)
47 \( 1 - 3.61T + 47T^{2} \)
53 \( 1 + 4.09T + 53T^{2} \)
59 \( 1 - 12.6T + 59T^{2} \)
61 \( 1 - 10.4iT - 61T^{2} \)
67 \( 1 + 9.43iT - 67T^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 - 6.66iT - 73T^{2} \)
79 \( 1 - 1.41iT - 79T^{2} \)
83 \( 1 - 0.543T + 83T^{2} \)
89 \( 1 - 0.554iT - 89T^{2} \)
97 \( 1 + 10.8iT - 97T^{2} \)
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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

−10.52725941699851393985361801178, −9.721007480135429067734607025295, −8.709961405462273815248260459380, −8.173343855923604916632624491194, −7.30697496163011732771012659885, −6.28746476179350772537779380817, −5.38143639480652192837944215847, −3.40030963283269115989687941772, −2.75096565894104182827875493047, −0.953702447414658018111453212768, 1.45527940591372810576326316697, 2.53335899746608145265995416723, 3.96858590985942636688055683474, 5.16580309750718699638249464647, 6.75675769060298425379762003309, 7.29459418441137993104152327899, 8.224516549763564848611928743861, 9.240079386768438403700079383652, 9.575920017911069407517132220476, 10.47509071669950531862445650615

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