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Properties

Label 2-588-7.2-c7-0-4
Degree 22
Conductor 588588
Sign 0.266−0.963i0.266 - 0.963i
Analytic cond. 183.682183.682
Root an. cond. 13.552913.5529
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−13.5 − 23.3i)3-s + (50 − 86.6i)5-s + (−364.5 + 631. i)9-s + (−1.38e3 − 2.40e3i)11-s + 3.29e3·13-s − 2.70e3·15-s + (2.95e3 + 5.10e3i)17-s + (3.32e3 − 5.75e3i)19-s + (−991 + 1.71e3i)23-s + (3.40e4 + 5.89e4i)25-s + 1.96e4·27-s − 2.08e5·29-s + (−5.88e4 − 1.02e5i)31-s + (−3.74e4 + 6.48e4i)33-s + (1.67e5 − 2.90e5i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.178 − 0.309i)5-s + (−0.166 + 0.288i)9-s + (−0.314 − 0.544i)11-s + 0.415·13-s − 0.206·15-s + (0.145 + 0.252i)17-s + (0.111 − 0.192i)19-s + (−0.0169 + 0.0294i)23-s + (0.435 + 0.755i)25-s + 0.192·27-s − 1.58·29-s + (−0.355 − 0.615i)31-s + (−0.181 + 0.314i)33-s + (0.544 − 0.943i)37-s + ⋯

Functional equation

Λ(s)=(588s/2 ΓC(s) L(s)=((0.266−0.963i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(588s/2 ΓC(s+7/2) L(s)=((0.266−0.963i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: 0.266−0.963i0.266 - 0.963i
Analytic conductor: 183.682183.682
Root analytic conductor: 13.552913.5529
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ588(373,⋅)\chi_{588} (373, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 588, ( :7/2), 0.266−0.963i)(2,\ 588,\ (\ :7/2),\ 0.266 - 0.963i)

Particular Values

L(4)L(4) ≈\approx 0.84010884180.8401088418
L(12)L(\frac12) ≈\approx 0.84010884180.8401088418
L(92)L(\frac{9}{2}) 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+(13.5+23.3i)T 1 + (13.5 + 23.3i)T
7 1 1
good5 1+(−50+86.6i)T+(−3.90e4−6.76e4i)T2 1 + (-50 + 86.6i)T + (-3.90e4 - 6.76e4i)T^{2}
11 1+(1.38e3+2.40e3i)T+(−9.74e6+1.68e7i)T2 1 + (1.38e3 + 2.40e3i)T + (-9.74e6 + 1.68e7i)T^{2}
13 1−3.29e3T+6.27e7T2 1 - 3.29e3T + 6.27e7T^{2}
17 1+(−2.95e3−5.10e3i)T+(−2.05e8+3.55e8i)T2 1 + (-2.95e3 - 5.10e3i)T + (-2.05e8 + 3.55e8i)T^{2}
19 1+(−3.32e3+5.75e3i)T+(−4.46e8−7.74e8i)T2 1 + (-3.32e3 + 5.75e3i)T + (-4.46e8 - 7.74e8i)T^{2}
23 1+(991−1.71e3i)T+(−1.70e9−2.94e9i)T2 1 + (991 - 1.71e3i)T + (-1.70e9 - 2.94e9i)T^{2}
29 1+2.08e5T+1.72e10T2 1 + 2.08e5T + 1.72e10T^{2}
31 1+(5.88e4+1.02e5i)T+(−1.37e10+2.38e10i)T2 1 + (5.88e4 + 1.02e5i)T + (-1.37e10 + 2.38e10i)T^{2}
37 1+(−1.67e5+2.90e5i)T+(−4.74e10−8.22e10i)T2 1 + (-1.67e5 + 2.90e5i)T + (-4.74e10 - 8.22e10i)T^{2}
41 1−2.65e5T+1.94e11T2 1 - 2.65e5T + 1.94e11T^{2}
43 1+9.32e4T+2.71e11T2 1 + 9.32e4T + 2.71e11T^{2}
47 1+(3.28e5−5.69e5i)T+(−2.53e11−4.38e11i)T2 1 + (3.28e5 - 5.69e5i)T + (-2.53e11 - 4.38e11i)T^{2}
53 1+(−3.04e5−5.27e5i)T+(−5.87e11+1.01e12i)T2 1 + (-3.04e5 - 5.27e5i)T + (-5.87e11 + 1.01e12i)T^{2}
59 1+(2.68e5+4.64e5i)T+(−1.24e12+2.15e12i)T2 1 + (2.68e5 + 4.64e5i)T + (-1.24e12 + 2.15e12i)T^{2}
61 1+(8.98e5−1.55e6i)T+(−1.57e12−2.72e12i)T2 1 + (8.98e5 - 1.55e6i)T + (-1.57e12 - 2.72e12i)T^{2}
67 1+(1.06e6+1.83e6i)T+(−3.03e12+5.24e12i)T2 1 + (1.06e6 + 1.83e6i)T + (-3.03e12 + 5.24e12i)T^{2}
71 1+1.19e6T+9.09e12T2 1 + 1.19e6T + 9.09e12T^{2}
73 1+(−5.28e5−9.14e5i)T+(−5.52e12+9.56e12i)T2 1 + (-5.28e5 - 9.14e5i)T + (-5.52e12 + 9.56e12i)T^{2}
79 1+(4.99e5−8.64e5i)T+(−9.60e12−1.66e13i)T2 1 + (4.99e5 - 8.64e5i)T + (-9.60e12 - 1.66e13i)T^{2}
83 1+3.89e6T+2.71e13T2 1 + 3.89e6T + 2.71e13T^{2}
89 1+(2.31e6−4.00e6i)T+(−2.21e13−3.83e13i)T2 1 + (2.31e6 - 4.00e6i)T + (-2.21e13 - 3.83e13i)T^{2}
97 1+1.52e7T+8.07e13T2 1 + 1.52e7T + 8.07e13T^{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

−9.625381915387839157045186062781, −8.885573298601572193733504077793, −7.900890635106704291208135095297, −7.16429055439546099825299757029, −5.96875205420502868933299406361, −5.47837886231782694960400297779, −4.21334224100799842892188565284, −3.06666647050749870565052171681, −1.85454728327641475132521290061, −0.891683237118341245798690430846, 0.18476175252631312872971162899, 1.55378111029448780272211417178, 2.75953644870299070314037985590, 3.79896731274207325923940704881, 4.81690428332309209732710504716, 5.70192651378745701886026809143, 6.64045127505009936824922357413, 7.57297566452261101417589658592, 8.604843249529118152959549252229, 9.551411826891306780185316463456

Graph of the ZZ-function along the critical line