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Properties

Label 2-588-147.2-c0-0-0
Degree 22
Conductor 588588
Sign 0.656+0.754i0.656 + 0.754i
Analytic cond. 0.2934500.293450
Root an. cond. 0.5417100.541710
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(588s/2 ΓC(s) L(s)=((0.656+0.754i) Λ‾(1−s)\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}
Λ(s)=(588s/2 ΓC(s) L(s)=((0.656+0.754i) Λ‾(1−s)\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: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: 0.656+0.754i0.656 + 0.754i
Analytic conductor: 0.2934500.293450
Root analytic conductor: 0.5417100.541710
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ588(149,⋅)\chi_{588} (149, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 588, ( :0), 0.656+0.754i)(2,\ 588,\ (\ :0),\ 0.656 + 0.754i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0587458021.058745802
L(12)L(\frac12) ≈\approx 1.0587458021.058745802
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(−0.365+0.930i)T 1 + (-0.365 + 0.930i)T
7 1+(−0.955−0.294i)T 1 + (-0.955 - 0.294i)T
good5 1+(−0.955+0.294i)T2 1 + (-0.955 + 0.294i)T^{2}
11 1+(−0.0747+0.997i)T2 1 + (-0.0747 + 0.997i)T^{2}
13 1+(0.162+0.712i)T+(−0.900+0.433i)T2 1 + (0.162 + 0.712i)T + (-0.900 + 0.433i)T^{2}
17 1+(0.988−0.149i)T2 1 + (0.988 - 0.149i)T^{2}
19 1+(0.955−1.65i)T+(−0.5−0.866i)T2 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2}
23 1+(0.988+0.149i)T2 1 + (0.988 + 0.149i)T^{2}
29 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
31 1+(0.826+1.43i)T+(−0.5+0.866i)T2 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2}
37 1+(−0.123+0.0841i)T+(0.365−0.930i)T2 1 + (-0.123 + 0.0841i)T + (0.365 - 0.930i)T^{2}
41 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
43 1+(0.914−1.14i)T+(−0.222−0.974i)T2 1 + (0.914 - 1.14i)T + (-0.222 - 0.974i)T^{2}
47 1+(−0.826−0.563i)T2 1 + (-0.826 - 0.563i)T^{2}
53 1+(−0.365−0.930i)T2 1 + (-0.365 - 0.930i)T^{2}
59 1+(−0.955−0.294i)T2 1 + (-0.955 - 0.294i)T^{2}
61 1+(1.48−1.01i)T+(0.365−0.930i)T2 1 + (1.48 - 1.01i)T + (0.365 - 0.930i)T^{2}
67 1+(−0.988−1.71i)T+(−0.5+0.866i)T2 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2}
71 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
73 1+(−0.142+0.0440i)T+(0.826−0.563i)T2 1 + (-0.142 + 0.0440i)T + (0.826 - 0.563i)T^{2}
79 1+(0.826−1.43i)T+(−0.5−0.866i)T2 1 + (0.826 - 1.43i)T + (-0.5 - 0.866i)T^{2}
83 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
89 1+(−0.0747−0.997i)T2 1 + (-0.0747 - 0.997i)T^{2}
97 1+0.445T+T2 1 + 0.445T + T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line