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Properties

Label 2-3040-3040.2469-c0-0-6
Degree 22
Conductor 30403040
Sign −0.773+0.634i-0.773 + 0.634i
Analytic cond. 1.517151.51715
Root an. cond. 1.231721.23172
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  + (−0.773 − 0.634i)2-s + (−0.0750 − 0.181i)3-s + (0.195 + 0.980i)4-s + (−0.923 − 0.382i)5-s + (−0.0569 + 0.187i)6-s + (0.471 − 0.881i)8-s + (0.679 − 0.679i)9-s + (0.471 + 0.881i)10-s + (0.425 − 1.02i)11-s + (0.162 − 0.108i)12-s + (−0.536 + 0.222i)13-s + 0.196i·15-s + (−0.923 + 0.382i)16-s + (−0.956 + 0.0942i)18-s + (0.923 − 0.382i)19-s + (0.195 − 0.980i)20-s + ⋯
L(s)  = 1  + (−0.773 − 0.634i)2-s + (−0.0750 − 0.181i)3-s + (0.195 + 0.980i)4-s + (−0.923 − 0.382i)5-s + (−0.0569 + 0.187i)6-s + (0.471 − 0.881i)8-s + (0.679 − 0.679i)9-s + (0.471 + 0.881i)10-s + (0.425 − 1.02i)11-s + (0.162 − 0.108i)12-s + (−0.536 + 0.222i)13-s + 0.196i·15-s + (−0.923 + 0.382i)16-s + (−0.956 + 0.0942i)18-s + (0.923 − 0.382i)19-s + (0.195 − 0.980i)20-s + ⋯

Functional equation

Λ(s)=(3040s/2 ΓC(s) L(s)=((−0.773+0.634i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3040s/2 ΓC(s) L(s)=((−0.773+0.634i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30403040    =    25⋅5⋅192^{5} \cdot 5 \cdot 19
Sign: −0.773+0.634i-0.773 + 0.634i
Analytic conductor: 1.517151.51715
Root analytic conductor: 1.231721.23172
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3040(2469,⋅)\chi_{3040} (2469, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3040, ( :0), −0.773+0.634i)(2,\ 3040,\ (\ :0),\ -0.773 + 0.634i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.61846663960.6184666396
L(12)L(\frac12) ≈\approx 0.61846663960.6184666396
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+(0.773+0.634i)T 1 + (0.773 + 0.634i)T
5 1+(0.923+0.382i)T 1 + (0.923 + 0.382i)T
19 1+(−0.923+0.382i)T 1 + (-0.923 + 0.382i)T
good3 1+(0.0750+0.181i)T+(−0.707+0.707i)T2 1 + (0.0750 + 0.181i)T + (-0.707 + 0.707i)T^{2}
7 1−iT2 1 - iT^{2}
11 1+(−0.425+1.02i)T+(−0.707−0.707i)T2 1 + (-0.425 + 1.02i)T + (-0.707 - 0.707i)T^{2}
13 1+(0.536−0.222i)T+(0.707−0.707i)T2 1 + (0.536 - 0.222i)T + (0.707 - 0.707i)T^{2}
17 1+T2 1 + T^{2}
23 1+iT2 1 + iT^{2}
29 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
31 1−T2 1 - T^{2}
37 1+(1.17+0.485i)T+(0.707+0.707i)T2 1 + (1.17 + 0.485i)T + (0.707 + 0.707i)T^{2}
41 1+iT2 1 + iT^{2}
43 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
47 1+T2 1 + T^{2}
53 1+(−0.761+1.83i)T+(−0.707−0.707i)T2 1 + (-0.761 + 1.83i)T + (-0.707 - 0.707i)T^{2}
59 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
61 1+(0.636+1.53i)T+(−0.707+0.707i)T2 1 + (0.636 + 1.53i)T + (-0.707 + 0.707i)T^{2}
67 1+(−0.674−1.62i)T+(−0.707+0.707i)T2 1 + (-0.674 - 1.62i)T + (-0.707 + 0.707i)T^{2}
71 1−iT2 1 - iT^{2}
73 1+iT2 1 + iT^{2}
79 1+T2 1 + T^{2}
83 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
89 1−iT2 1 - iT^{2}
97 1+1.91T+T2 1 + 1.91T + 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

−8.616859716811468410627119277363, −8.062423296448466755143705682347, −7.11608615303725494339779635000, −6.82608175657669909926269479836, −5.51026854067522384188121339570, −4.40454437397199797569210489640, −3.68782658223215972166364959834, −3.00039716687579371709103869834, −1.56614756203313262153902540699, −0.55740925441364921504112373537, 1.38411054141067054627073567722, 2.52290824605684265291311444625, 3.84408869033054187005796502001, 4.70989379598154054300691940479, 5.32064662661404208474620166403, 6.48072661250551358605524981584, 7.29660049694875678133209034738, 7.47398474399960117105401500739, 8.289687694605314470214986650450, 9.171318352709536925050673513126

Graph of the ZZ-function along the critical line