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Properties

Label 2-588-147.2-c0-0-0
Degree $2$
Conductor $588$
Sign $0.656 + 0.754i$
Analytic cond. $0.293450$
Root an. cond. $0.541710$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.365 − 0.930i)3-s + (0.955 + 0.294i)7-s + (−0.733 − 0.680i)9-s + (−0.162 − 0.712i)13-s + (−0.955 + 1.65i)19-s + (0.623 − 0.781i)21-s + (0.955 − 0.294i)25-s + (−0.900 + 0.433i)27-s + (−0.826 − 1.43i)31-s + (0.123 − 0.0841i)37-s + (−0.722 − 0.108i)39-s + (−0.914 + 1.14i)43-s + (0.826 + 0.563i)49-s + (1.19 + 1.49i)57-s + (−1.48 + 1.01i)61-s + ⋯
L(s)  = 1  + (0.365 − 0.930i)3-s + (0.955 + 0.294i)7-s + (−0.733 − 0.680i)9-s + (−0.162 − 0.712i)13-s + (−0.955 + 1.65i)19-s + (0.623 − 0.781i)21-s + (0.955 − 0.294i)25-s + (−0.900 + 0.433i)27-s + (−0.826 − 1.43i)31-s + (0.123 − 0.0841i)37-s + (−0.722 − 0.108i)39-s + (−0.914 + 1.14i)43-s + (0.826 + 0.563i)49-s + (1.19 + 1.49i)57-s + (−1.48 + 1.01i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.656 + 0.754i$
Analytic conductor: \(0.293450\)
Root analytic conductor: \(0.541710\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :0),\ 0.656 + 0.754i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.058745802\)
\(L(\frac12)\) \(\approx\) \(1.058745802\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.365 + 0.930i)T \)
7 \( 1 + (-0.955 - 0.294i)T \)
good5 \( 1 + (-0.955 + 0.294i)T^{2} \)
11 \( 1 + (-0.0747 + 0.997i)T^{2} \)
13 \( 1 + (0.162 + 0.712i)T + (-0.900 + 0.433i)T^{2} \)
17 \( 1 + (0.988 - 0.149i)T^{2} \)
19 \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.988 + 0.149i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.123 + 0.0841i)T + (0.365 - 0.930i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (0.914 - 1.14i)T + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (-0.826 - 0.563i)T^{2} \)
53 \( 1 + (-0.365 - 0.930i)T^{2} \)
59 \( 1 + (-0.955 - 0.294i)T^{2} \)
61 \( 1 + (1.48 - 1.01i)T + (0.365 - 0.930i)T^{2} \)
67 \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.142 + 0.0440i)T + (0.826 - 0.563i)T^{2} \)
79 \( 1 + (0.826 - 1.43i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.0747 - 0.997i)T^{2} \)
97 \( 1 + 0.445T + T^{2} \)
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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

−10.92203632131786625418748884954, −9.904766474933104848514331809085, −8.710344682811374315385849237885, −8.122516548400894825485095697460, −7.44032358348053150852922398315, −6.24818407639563623620885601400, −5.45819912085152944696962286714, −4.08989798739970360267810482275, −2.69671423820908542755360543616, −1.55666845176746133258904324933, 2.06715429261956956760919668483, 3.41845638561531586576518534399, 4.65024968572362047871172068240, 5.04234984429616369872013149462, 6.60176057751751878237849297667, 7.56650775776052910119368067032, 8.760332554025244268183448785121, 9.014108037172229645821252531473, 10.29958024115877648227223473393, 10.93457115634280473634913611373

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