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Properties

Label 2-588-12.11-c1-0-31
Degree $2$
Conductor $588$
Sign $-i$
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  + 1.41·2-s + 1.73i·3-s + 2.00·4-s + 2.44i·5-s + 2.44i·6-s + 2.82·8-s − 2.99·9-s + 3.46i·10-s + 1.41·11-s + 3.46i·12-s − 4.24·15-s + 4.00·16-s − 7.34i·17-s − 4.24·18-s + 6.92i·19-s + 4.89i·20-s + ⋯
L(s)  = 1  + 1.00·2-s + 0.999i·3-s + 1.00·4-s + 1.09i·5-s + 0.999i·6-s + 1.00·8-s − 0.999·9-s + 1.09i·10-s + 0.426·11-s + 1.00i·12-s − 1.09·15-s + 1.00·16-s − 1.78i·17-s − 0.999·18-s + 1.58i·19-s + 1.09i·20-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.90866 + 1.90866i\)
\(L(\frac12)\) \(\approx\) \(1.90866 + 1.90866i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 - 1.73iT \)
7 \( 1 \)
good5 \( 1 - 2.44iT - 5T^{2} \)
11 \( 1 - 1.41T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 7.34iT - 17T^{2} \)
19 \( 1 - 6.92iT - 19T^{2} \)
23 \( 1 + 7.07T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + 12.2iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 15.5T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 2.44iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−10.95509561843821468649148461601, −10.23360949161724708978534937655, −9.525225332282960596936064902606, −8.099466575792251826132653316832, −7.11422190130482753024056888433, −6.15972532562694323325234725952, −5.34238552162109844916914362071, −4.16714780119131388009677033715, −3.39718766548221830009475683536, −2.39030839762937116796150187247, 1.23397632067786385446428798108, 2.41067213216291527618734691998, 3.91398293944621002883742720701, 4.88928075364511525773400536876, 6.00090107098210603429995128677, 6.54892801079852363739379340599, 7.81385800512075441609086554650, 8.387623221404211237741243877106, 9.534840604533915201080312237256, 10.88587514871838538762952254697

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