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Properties

Label 2-588-12.11-c1-0-5
Degree $2$
Conductor $588$
Sign $0.228 - 0.973i$
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  + (−0.446 − 1.34i)2-s + (−1.32 + 1.11i)3-s + (−1.60 + 1.19i)4-s − 0.803i·5-s + (2.08 + 1.28i)6-s + (2.32 + 1.61i)8-s + (0.523 − 2.95i)9-s + (−1.07 + 0.358i)10-s + 2.34·11-s + (0.790 − 3.37i)12-s − 5.26·13-s + (0.893 + 1.06i)15-s + (1.12 − 3.83i)16-s − 1.18i·17-s + (−4.19 + 0.616i)18-s + 7.12i·19-s + ⋯
L(s)  = 1  + (−0.315 − 0.948i)2-s + (−0.766 + 0.642i)3-s + (−0.800 + 0.599i)4-s − 0.359i·5-s + (0.851 + 0.524i)6-s + (0.821 + 0.569i)8-s + (0.174 − 0.984i)9-s + (−0.340 + 0.113i)10-s + 0.707·11-s + (0.228 − 0.973i)12-s − 1.46·13-s + (0.230 + 0.275i)15-s + (0.281 − 0.959i)16-s − 0.287i·17-s + (−0.989 + 0.145i)18-s + 1.63i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.228 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.228 - 0.973i$
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: \((2,\ 588,\ (\ :1/2),\ 0.228 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.359448 + 0.284922i\)
\(L(\frac12)\) \(\approx\) \(0.359448 + 0.284922i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.446 + 1.34i)T \)
3 \( 1 + (1.32 - 1.11i)T \)
7 \( 1 \)
good5 \( 1 + 0.803iT - 5T^{2} \)
11 \( 1 - 2.34T + 11T^{2} \)
13 \( 1 + 5.26T + 13T^{2} \)
17 \( 1 + 1.18iT - 17T^{2} \)
19 \( 1 - 7.12iT - 19T^{2} \)
23 \( 1 + 7.88T + 23T^{2} \)
29 \( 1 - 4.23iT - 29T^{2} \)
31 \( 1 - 4.89iT - 31T^{2} \)
37 \( 1 - 1.04T + 37T^{2} \)
41 \( 1 - 7.16iT - 41T^{2} \)
43 \( 1 - 7.94iT - 43T^{2} \)
47 \( 1 + 6.09T + 47T^{2} \)
53 \( 1 - 8.72iT - 53T^{2} \)
59 \( 1 + 0.662T + 59T^{2} \)
61 \( 1 + 0.958T + 61T^{2} \)
67 \( 1 + 8.42iT - 67T^{2} \)
71 \( 1 + 9.67T + 71T^{2} \)
73 \( 1 - 1.41T + 73T^{2} \)
79 \( 1 - 6.92iT - 79T^{2} \)
83 \( 1 + 5.18T + 83T^{2} \)
89 \( 1 + 16.3iT - 89T^{2} \)
97 \( 1 - 4.37T + 97T^{2} \)
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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

−10.79450135095241804001829577754, −9.969768705055979262417637964151, −9.564367558813524072451294796840, −8.538095280564644835735344928387, −7.48691182672173073998479977051, −6.17507842281887323547770970698, −4.99415472600793530729833309070, −4.30339840870110614531462824256, −3.17401869765301374665551479141, −1.46626589557926047902637457486, 0.32950929117528673586148030274, 2.18094735133590100415923417494, 4.26658733260639331407617501724, 5.19437082264126226708951353752, 6.18108102667769609045801186904, 6.94821444274531005832676854582, 7.52093085337867004134309022464, 8.558088011772713703862603308311, 9.644924929707522241229054511744, 10.35256092307345821177689393303

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