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Properties

Label 2-580-580.419-c0-0-1
Degree 22
Conductor 580580
Sign 0.441−0.897i0.441 - 0.897i
Analytic cond. 0.2894570.289457
Root an. cond. 0.5380120.538012
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  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.623 + 0.781i)5-s + (0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.900 + 0.433i)10-s + (1.52 + 0.347i)13-s + (−0.222 + 0.974i)16-s − 1.80·17-s + (0.222 − 0.974i)18-s − 20-s + (−0.222 − 0.974i)25-s + (1.22 + 0.974i)26-s + (0.222 − 0.974i)29-s + (−0.623 + 0.781i)32-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.623 + 0.781i)5-s + (0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.900 + 0.433i)10-s + (1.52 + 0.347i)13-s + (−0.222 + 0.974i)16-s − 1.80·17-s + (0.222 − 0.974i)18-s − 20-s + (−0.222 − 0.974i)25-s + (1.22 + 0.974i)26-s + (0.222 − 0.974i)29-s + (−0.623 + 0.781i)32-s + ⋯

Functional equation

Λ(s)=(580s/2 ΓC(s) L(s)=((0.441−0.897i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(580s/2 ΓC(s) L(s)=((0.441−0.897i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 580580    =    22⋅5⋅292^{2} \cdot 5 \cdot 29
Sign: 0.441−0.897i0.441 - 0.897i
Analytic conductor: 0.2894570.289457
Root analytic conductor: 0.5380120.538012
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ580(419,⋅)\chi_{580} (419, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 580, ( :0), 0.441−0.897i)(2,\ 580,\ (\ :0),\ 0.441 - 0.897i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3625999331.362599933
L(12)L(\frac12) ≈\approx 1.3625999331.362599933
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+(−0.900−0.433i)T 1 + (-0.900 - 0.433i)T
5 1+(0.623−0.781i)T 1 + (0.623 - 0.781i)T
29 1+(−0.222+0.974i)T 1 + (-0.222 + 0.974i)T
good3 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
7 1+(−0.222−0.974i)T2 1 + (-0.222 - 0.974i)T^{2}
11 1+(−0.900−0.433i)T2 1 + (-0.900 - 0.433i)T^{2}
13 1+(−1.52−0.347i)T+(0.900+0.433i)T2 1 + (-1.52 - 0.347i)T + (0.900 + 0.433i)T^{2}
17 1+1.80T+T2 1 + 1.80T + T^{2}
19 1+(−0.222+0.974i)T2 1 + (-0.222 + 0.974i)T^{2}
23 1+(0.623−0.781i)T2 1 + (0.623 - 0.781i)T^{2}
31 1+(0.623+0.781i)T2 1 + (0.623 + 0.781i)T^{2}
37 1+(0.277+1.21i)T+(−0.900+0.433i)T2 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2}
41 1+1.56iT−T2 1 + 1.56iT - T^{2}
43 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
47 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
53 1+(−0.376+0.781i)T+(−0.623−0.781i)T2 1 + (-0.376 + 0.781i)T + (-0.623 - 0.781i)T^{2}
59 1−T2 1 - T^{2}
61 1+(−0.678−0.541i)T+(0.222+0.974i)T2 1 + (-0.678 - 0.541i)T + (0.222 + 0.974i)T^{2}
67 1+(−0.900+0.433i)T2 1 + (-0.900 + 0.433i)T^{2}
71 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
73 1+(1.62−0.781i)T+(0.623−0.781i)T2 1 + (1.62 - 0.781i)T + (0.623 - 0.781i)T^{2}
79 1+(−0.900+0.433i)T2 1 + (-0.900 + 0.433i)T^{2}
83 1+(−0.222+0.974i)T2 1 + (-0.222 + 0.974i)T^{2}
89 1+(0.846−1.75i)T+(−0.623−0.781i)T2 1 + (0.846 - 1.75i)T + (-0.623 - 0.781i)T^{2}
97 1+(0.777+0.974i)T+(−0.222+0.974i)T2 1 + (0.777 + 0.974i)T + (-0.222 + 0.974i)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

−11.27873439430746192122302912146, −10.62118702019868772747839513774, −9.024579680535958772797008644967, −8.378320428893708805035660713849, −7.16454371891961185441362571337, −6.53334019678693129820153839089, −5.81537489412941952641311170558, −4.18998801268479225140940596260, −3.73171193255862131468476489472, −2.44541356668335301913385293995, 1.58495374033259793645313059458, 3.08542149507336186641921875867, 4.26101103228687722074456990902, 4.94891266858901709766676690871, 5.99338433081824304233099100540, 7.02138907511285923519816626893, 8.264112197088242833604015494002, 8.876312980051317415787712305591, 10.23078249620950821145710743450, 11.14051118176512432148844924560

Graph of the ZZ-function along the critical line