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Properties

Label 2-588-12.11-c1-0-38
Degree $2$
Conductor $588$
Sign $0.815 + 0.579i$
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  + (0.349 − 1.37i)2-s + (1.72 + 0.203i)3-s + (−1.75 − 0.958i)4-s + 2.27i·5-s + (0.880 − 2.28i)6-s + (−1.92 + 2.06i)8-s + (2.91 + 0.699i)9-s + (3.12 + 0.796i)10-s + 4.31·11-s + (−2.82 − 2.00i)12-s + 0.406·13-s + (−0.463 + 3.91i)15-s + (2.16 + 3.36i)16-s − 4.31i·17-s + (1.97 − 3.75i)18-s + 5.42i·19-s + ⋯
L(s)  = 1  + (0.247 − 0.968i)2-s + (0.993 + 0.117i)3-s + (−0.877 − 0.479i)4-s + 1.01i·5-s + (0.359 − 0.933i)6-s + (−0.681 + 0.731i)8-s + (0.972 + 0.233i)9-s + (0.986 + 0.251i)10-s + 1.30·11-s + (−0.815 − 0.579i)12-s + 0.112·13-s + (−0.119 + 1.01i)15-s + (0.540 + 0.841i)16-s − 1.04i·17-s + (0.466 − 0.884i)18-s + 1.24i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.815 + 0.579i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.815 + 0.579i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13436 - 0.680830i\)
\(L(\frac12)\) \(\approx\) \(2.13436 - 0.680830i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.349 + 1.37i)T \)
3 \( 1 + (-1.72 - 0.203i)T \)
7 \( 1 \)
good5 \( 1 - 2.27iT - 5T^{2} \)
11 \( 1 - 4.31T + 11T^{2} \)
13 \( 1 - 0.406T + 13T^{2} \)
17 \( 1 + 4.31iT - 17T^{2} \)
19 \( 1 - 5.42iT - 19T^{2} \)
23 \( 1 - 1.16T + 23T^{2} \)
29 \( 1 + 3.72iT - 29T^{2} \)
31 \( 1 + 5.83iT - 31T^{2} \)
37 \( 1 + 7.83T + 37T^{2} \)
41 \( 1 + 2.56iT - 41T^{2} \)
43 \( 1 - 11.0iT - 43T^{2} \)
47 \( 1 + 2.32T + 47T^{2} \)
53 \( 1 + 5.48iT - 53T^{2} \)
59 \( 1 + 3.91T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 - 7.83iT - 67T^{2} \)
71 \( 1 + 4.88T + 71T^{2} \)
73 \( 1 + 2.81T + 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 + 0.641T + 83T^{2} \)
89 \( 1 + 13.9iT - 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−10.54349976976227331074499873651, −9.762181866019533803153255336207, −9.176339851774088745598454967052, −8.158262422993317181631928034244, −7.08902258601889418357347205499, −6.04472830364021566868732113957, −4.52577427320016148868006415319, −3.61419449413602469259667935794, −2.84317676783894268128165290217, −1.61837036147840429504467658809, 1.35986859538439659027035822726, 3.33057168299526244410134626465, 4.26527045050063771348868468254, 5.12568252549115280773216581785, 6.48825294609614577299700437449, 7.15906243284187840032295962444, 8.307891546971475185709715163362, 8.943880499430205307092607843282, 9.225868096267554856857717998068, 10.55860899285983832055505693270

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