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Properties

Label 2-588-21.20-c5-0-11
Degree 22
Conductor 588588
Sign −0.591−0.805i-0.591 - 0.805i
Analytic cond. 94.305694.3056
Root an. cond. 9.711119.71111
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−10.9 + 11.0i)3-s + 30.7·5-s + (−2.58 − 242. i)9-s + 13.8i·11-s − 860. i·13-s + (−337. + 341. i)15-s − 1.17e3·17-s + 681. i·19-s − 880. i·23-s − 2.17e3·25-s + (2.72e3 + 2.63e3i)27-s − 3.78e3i·29-s + 6.09e3i·31-s + (−153. − 152. i)33-s − 2.43e3·37-s + ⋯
L(s)  = 1  + (−0.703 + 0.710i)3-s + 0.550·5-s + (−0.0106 − 0.999i)9-s + 0.0346i·11-s − 1.41i·13-s + (−0.387 + 0.391i)15-s − 0.984·17-s + 0.433i·19-s − 0.347i·23-s − 0.696·25-s + (0.718 + 0.695i)27-s − 0.835i·29-s + 1.13i·31-s + (−0.0246 − 0.0243i)33-s − 0.292·37-s + ⋯

Functional equation

Λ(s)=(588s/2 ΓC(s) L(s)=((−0.591−0.805i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 - 0.805i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(588s/2 ΓC(s+5/2) L(s)=((−0.591−0.805i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.591 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: −0.591−0.805i-0.591 - 0.805i
Analytic conductor: 94.305694.3056
Root analytic conductor: 9.711119.71111
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ588(293,⋅)\chi_{588} (293, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 588, ( :5/2), −0.591−0.805i)(2,\ 588,\ (\ :5/2),\ -0.591 - 0.805i)

Particular Values

L(3)L(3) ≈\approx 0.86093939120.8609393912
L(12)L(\frac12) ≈\approx 0.86093939120.8609393912
L(72)L(\frac{7}{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+(10.9−11.0i)T 1 + (10.9 - 11.0i)T
7 1 1
good5 1−30.7T+3.12e3T2 1 - 30.7T + 3.12e3T^{2}
11 1−13.8iT−1.61e5T2 1 - 13.8iT - 1.61e5T^{2}
13 1+860.iT−3.71e5T2 1 + 860. iT - 3.71e5T^{2}
17 1+1.17e3T+1.41e6T2 1 + 1.17e3T + 1.41e6T^{2}
19 1−681.iT−2.47e6T2 1 - 681. iT - 2.47e6T^{2}
23 1+880.iT−6.43e6T2 1 + 880. iT - 6.43e6T^{2}
29 1+3.78e3iT−2.05e7T2 1 + 3.78e3iT - 2.05e7T^{2}
31 1−6.09e3iT−2.86e7T2 1 - 6.09e3iT - 2.86e7T^{2}
37 1+2.43e3T+6.93e7T2 1 + 2.43e3T + 6.93e7T^{2}
41 1−8.74e3T+1.15e8T2 1 - 8.74e3T + 1.15e8T^{2}
43 1+4.82e3T+1.47e8T2 1 + 4.82e3T + 1.47e8T^{2}
47 1−2.26e4T+2.29e8T2 1 - 2.26e4T + 2.29e8T^{2}
53 1−3.25e4iT−4.18e8T2 1 - 3.25e4iT - 4.18e8T^{2}
59 1+5.68e3T+7.14e8T2 1 + 5.68e3T + 7.14e8T^{2}
61 1+2.59e3iT−8.44e8T2 1 + 2.59e3iT - 8.44e8T^{2}
67 1+1.75e4T+1.35e9T2 1 + 1.75e4T + 1.35e9T^{2}
71 1−3.75e4iT−1.80e9T2 1 - 3.75e4iT - 1.80e9T^{2}
73 1+5.62e3iT−2.07e9T2 1 + 5.62e3iT - 2.07e9T^{2}
79 1+3.55e3T+3.07e9T2 1 + 3.55e3T + 3.07e9T^{2}
83 1−1.12e5T+3.93e9T2 1 - 1.12e5T + 3.93e9T^{2}
89 1−4.94e3T+5.58e9T2 1 - 4.94e3T + 5.58e9T^{2}
97 1−9.37e4iT−8.58e9T2 1 - 9.37e4iT - 8.58e9T^{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

−10.39607371624377691932867078697, −9.500268123002302675340877659598, −8.659176601540254237672932818793, −7.52137354423008206772920795680, −6.30472666669312363207989400921, −5.69599189947823478982020106523, −4.76667590404267728668928568506, −3.72213364001292556529880365595, −2.49455562517168638319220624221, −0.946234843916634256259961719873, 0.23923915975400780849660285361, 1.62505323595414221299209749344, 2.36153583604324585569624415056, 4.10417057525675308202396111787, 5.11176450729189502917827738409, 6.11643343598016610781745979895, 6.77511812525332803045632315781, 7.61497253753000996848054084092, 8.815910473928303456507536742331, 9.556203968361779455244994979674

Graph of the ZZ-function along the critical line