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Properties

Label 2-588-12.11-c1-0-27
Degree $2$
Conductor $588$
Sign $0.665 - 0.746i$
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  + (−0.225 + 1.39i)2-s + (0.687 − 1.58i)3-s + (−1.89 − 0.629i)4-s + 2.07i·5-s + (2.06 + 1.31i)6-s + (1.30 − 2.50i)8-s + (−2.05 − 2.18i)9-s + (−2.89 − 0.467i)10-s + 3.98·11-s + (−2.30 + 2.58i)12-s + 1.30·13-s + (3.30 + 1.42i)15-s + (3.20 + 2.38i)16-s + 2.94i·17-s + (3.51 − 2.37i)18-s + 1.09i·19-s + ⋯
L(s)  = 1  + (−0.159 + 0.987i)2-s + (0.397 − 0.917i)3-s + (−0.949 − 0.314i)4-s + 0.928i·5-s + (0.842 + 0.538i)6-s + (0.461 − 0.887i)8-s + (−0.684 − 0.729i)9-s + (−0.916 − 0.147i)10-s + 1.20·11-s + (−0.665 + 0.746i)12-s + 0.363·13-s + (0.852 + 0.368i)15-s + (0.802 + 0.597i)16-s + 0.713i·17-s + (0.828 − 0.559i)18-s + 0.251i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.665 - 0.746i$
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: \((2,\ 588,\ (\ :1/2),\ 0.665 - 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36211 + 0.610218i\)
\(L(\frac12)\) \(\approx\) \(1.36211 + 0.610218i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.225 - 1.39i)T \)
3 \( 1 + (-0.687 + 1.58i)T \)
7 \( 1 \)
good5 \( 1 - 2.07iT - 5T^{2} \)
11 \( 1 - 3.98T + 11T^{2} \)
13 \( 1 - 1.30T + 13T^{2} \)
17 \( 1 - 2.94iT - 17T^{2} \)
19 \( 1 - 1.09iT - 19T^{2} \)
23 \( 1 - 7.50T + 23T^{2} \)
29 \( 1 + 0.865iT - 29T^{2} \)
31 \( 1 + 3.68iT - 31T^{2} \)
37 \( 1 - 4.17T + 37T^{2} \)
41 \( 1 - 7.01iT - 41T^{2} \)
43 \( 1 - 4.27iT - 43T^{2} \)
47 \( 1 - 7.50T + 47T^{2} \)
53 \( 1 + 4.94iT - 53T^{2} \)
59 \( 1 - 4.88T + 59T^{2} \)
61 \( 1 + 4.10T + 61T^{2} \)
67 \( 1 + 2.42iT - 67T^{2} \)
71 \( 1 - 0.901T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 + 12.4T + 83T^{2} \)
89 \( 1 - 8.52iT - 89T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−10.79053043881244509911555872197, −9.599839951501470788067478807701, −8.854746518512615169323816175957, −8.014248157733384303240741326473, −7.08636653516786645246498734846, −6.53793880148059982634537897612, −5.80983328079861665692885916192, −4.18363940427748196425397475278, −3.06070883362744523026642109375, −1.28383194950219184960404184037, 1.13306658624469536015575344761, 2.76727737874650459432038790991, 3.86955228064073691563334146063, 4.66825328935398338587440993456, 5.49521131470839009339494896398, 7.22351501145316154311514215512, 8.665033846063018283422518419135, 8.899128196136155198336540103192, 9.572966480720893767944921875885, 10.60227921729915010542649678978

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