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Properties

Label 2-580-580.3-c0-0-0
Degree 22
Conductor 580580
Sign 0.973+0.227i0.973 + 0.227i
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  + (−0.974 + 0.222i)2-s + (0.900 − 0.433i)4-s + (0.433 − 0.900i)5-s + (−0.781 + 0.623i)8-s + (0.623 + 0.781i)9-s + (−0.222 + 0.974i)10-s + (0.0739 + 0.656i)13-s + (0.623 − 0.781i)16-s − 1.94i·17-s + (−0.781 − 0.623i)18-s − i·20-s + (−0.623 − 0.781i)25-s + (−0.218 − 0.623i)26-s + (0.781 + 0.623i)29-s + (−0.433 + 0.900i)32-s + ⋯
L(s)  = 1  + (−0.974 + 0.222i)2-s + (0.900 − 0.433i)4-s + (0.433 − 0.900i)5-s + (−0.781 + 0.623i)8-s + (0.623 + 0.781i)9-s + (−0.222 + 0.974i)10-s + (0.0739 + 0.656i)13-s + (0.623 − 0.781i)16-s − 1.94i·17-s + (−0.781 − 0.623i)18-s − i·20-s + (−0.623 − 0.781i)25-s + (−0.218 − 0.623i)26-s + (0.781 + 0.623i)29-s + (−0.433 + 0.900i)32-s + ⋯

Functional equation

Λ(s)=(580s/2 ΓC(s) L(s)=((0.973+0.227i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(580s/2 ΓC(s) L(s)=((0.973+0.227i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 580580    =    22⋅5⋅292^{2} \cdot 5 \cdot 29
Sign: 0.973+0.227i0.973 + 0.227i
Analytic conductor: 0.2894570.289457
Root analytic conductor: 0.5380120.538012
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ580(3,⋅)\chi_{580} (3, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 580, ( :0), 0.973+0.227i)(2,\ 580,\ (\ :0),\ 0.973 + 0.227i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.67586717240.6758671724
L(12)L(\frac12) ≈\approx 0.67586717240.6758671724
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+(0.974−0.222i)T 1 + (0.974 - 0.222i)T
5 1+(−0.433+0.900i)T 1 + (-0.433 + 0.900i)T
29 1+(−0.781−0.623i)T 1 + (-0.781 - 0.623i)T
good3 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
7 1+(−0.781+0.623i)T2 1 + (-0.781 + 0.623i)T^{2}
11 1+(0.974−0.222i)T2 1 + (0.974 - 0.222i)T^{2}
13 1+(−0.0739−0.656i)T+(−0.974+0.222i)T2 1 + (-0.0739 - 0.656i)T + (-0.974 + 0.222i)T^{2}
17 1+1.94iT−T2 1 + 1.94iT - T^{2}
19 1+(−0.781−0.623i)T2 1 + (-0.781 - 0.623i)T^{2}
23 1+(−0.433+0.900i)T2 1 + (-0.433 + 0.900i)T^{2}
31 1+(−0.433−0.900i)T2 1 + (-0.433 - 0.900i)T^{2}
37 1+(−1.12−1.40i)T+(−0.222+0.974i)T2 1 + (-1.12 - 1.40i)T + (-0.222 + 0.974i)T^{2}
41 1+(1.33+1.33i)T+iT2 1 + (1.33 + 1.33i)T + iT^{2}
43 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
47 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
53 1+(0.566−0.900i)T+(−0.433−0.900i)T2 1 + (0.566 - 0.900i)T + (-0.433 - 0.900i)T^{2}
59 1+T2 1 + T^{2}
61 1+(1.59−0.559i)T+(0.781−0.623i)T2 1 + (1.59 - 0.559i)T + (0.781 - 0.623i)T^{2}
67 1+(0.974+0.222i)T2 1 + (0.974 + 0.222i)T^{2}
71 1+(−0.222−0.974i)T2 1 + (-0.222 - 0.974i)T^{2}
73 1+(0.433+0.0990i)T+(0.900+0.433i)T2 1 + (0.433 + 0.0990i)T + (0.900 + 0.433i)T^{2}
79 1+(0.974+0.222i)T2 1 + (0.974 + 0.222i)T^{2}
83 1+(−0.781−0.623i)T2 1 + (-0.781 - 0.623i)T^{2}
89 1+(1.05−1.68i)T+(−0.433−0.900i)T2 1 + (1.05 - 1.68i)T + (-0.433 - 0.900i)T^{2}
97 1+(0.781−0.376i)T+(0.623−0.781i)T2 1 + (0.781 - 0.376i)T + (0.623 - 0.781i)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

−10.64593130410493557221926003843, −9.818157109211567959923238594485, −9.191259372817523253695266683254, −8.380114174242819171004951993073, −7.42274457321616635425387424862, −6.63789983539076172326376544139, −5.37406339500333712396953801046, −4.61390382933098271482369573623, −2.61897881850716786709465327375, −1.35833975393823302682912603802, 1.61442408924215957215912133969, 2.96653439068943675524219499991, 3.97785223895348764154886052324, 5.99354848338282178708247553256, 6.46903040996226286106166433456, 7.52671728901673121945235712072, 8.330783441536245466293252554503, 9.402563101604943090062803475103, 10.15423809263998371207669537349, 10.64835153304434470130885556805

Graph of the ZZ-function along the critical line