↑

Properties

Label 2-3040-3040.1709-c0-0-2
Degree 22
Conductor 30403040
Sign 0.773+0.634i0.773 + 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

Related objects

Downloads

Learn more

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+0.634i0.773 + 0.634i
Analytic conductor: 1.517151.51715
Root analytic conductor: 1.231721.23172
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3040(1709,⋅)\chi_{3040} (1709, \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 1.8459932961.845993296
L(12)L(\frac12) ≈\approx 1.8459932961.845993296
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}
show more
show less
   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

−8.896127257752942838675040280903, −7.87030074479517008780602808164, −7.20246832666292095857423701645, −6.64948063028317014933802982216, −5.61519154424590565774996071348, −4.64156802202322436984657425789, −4.13420731376855595334571459664, −3.31777758352774097978827016384, −2.27121978707050885142336920998, −1.29713672214449542396498264868, 1.10350572633682419028023088950, 3.03071731232576440405789048946, 3.53835155394797628140168419133, 4.32578584826370271905905571975, 4.99876452004165507298958685874, 6.06489314151319316284276395283, 6.54584088578353128324416717769, 7.63734212463274084185445557611, 7.904210548537236384591311149227, 9.031342643543614067317985610241

Graph of the ZZ-function along the critical line