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Properties

Label 2-588-147.23-c0-0-0
Degree $2$
Conductor $588$
Sign $0.871 + 0.490i$
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  + (0.955 − 0.294i)3-s + (0.0747 − 0.997i)7-s + (0.826 − 0.563i)9-s + (−1.72 + 0.829i)13-s + (−0.0747 + 0.129i)19-s + (−0.222 − 0.974i)21-s + (0.0747 + 0.997i)25-s + (0.623 − 0.781i)27-s + (0.988 + 1.71i)31-s + (−0.722 + 0.108i)37-s + (−1.40 + 1.29i)39-s + (−0.367 − 1.61i)43-s + (−0.988 − 0.149i)49-s + (−0.0332 + 0.145i)57-s + (−1.23 + 0.185i)61-s + ⋯
L(s)  = 1  + (0.955 − 0.294i)3-s + (0.0747 − 0.997i)7-s + (0.826 − 0.563i)9-s + (−1.72 + 0.829i)13-s + (−0.0747 + 0.129i)19-s + (−0.222 − 0.974i)21-s + (0.0747 + 0.997i)25-s + (0.623 − 0.781i)27-s + (0.988 + 1.71i)31-s + (−0.722 + 0.108i)37-s + (−1.40 + 1.29i)39-s + (−0.367 − 1.61i)43-s + (−0.988 − 0.149i)49-s + (−0.0332 + 0.145i)57-s + (−1.23 + 0.185i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.871 + 0.490i$
Analytic conductor: \(0.293450\)
Root analytic conductor: \(0.541710\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :0),\ 0.871 + 0.490i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.180895374\)
\(L(\frac12)\) \(\approx\) \(1.180895374\)
\(L(1)\) 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 \( 1 + (-0.955 + 0.294i)T \)
7 \( 1 + (-0.0747 + 0.997i)T \)
good5 \( 1 + (-0.0747 - 0.997i)T^{2} \)
11 \( 1 + (-0.365 - 0.930i)T^{2} \)
13 \( 1 + (1.72 - 0.829i)T + (0.623 - 0.781i)T^{2} \)
17 \( 1 + (0.733 + 0.680i)T^{2} \)
19 \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.733 - 0.680i)T^{2} \)
29 \( 1 + (0.222 - 0.974i)T^{2} \)
31 \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.722 - 0.108i)T + (0.955 - 0.294i)T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.367 + 1.61i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.988 + 0.149i)T^{2} \)
53 \( 1 + (-0.955 - 0.294i)T^{2} \)
59 \( 1 + (-0.0747 + 0.997i)T^{2} \)
61 \( 1 + (1.23 - 0.185i)T + (0.955 - 0.294i)T^{2} \)
67 \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.0546 - 0.728i)T + (-0.988 + 0.149i)T^{2} \)
79 \( 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.365 + 0.930i)T^{2} \)
97 \( 1 + 1.80T + T^{2} \)
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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

−10.59266794104363523535901504593, −9.901336312715754622516277214113, −9.101983417021709464379179872515, −8.149942581575917872035370934960, −7.13093244527214949503733897023, −6.87950073630855021649770342849, −5.06381654486377712037030853294, −4.12137799058218943481813301017, −2.99851932164108026702505154775, −1.67903677522444305254266723655, 2.26525933591875023635767300839, 2.95331654866925326455233834708, 4.43978048812425102550902004829, 5.26957849517628938219726786245, 6.53160259441540444603463224388, 7.80213804058705718923847174302, 8.208362343833780376133810583147, 9.417986382029737363457824390255, 9.788237941107163711228820923545, 10.81108296844765077789176212965

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