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Properties

Label 2-3040-3040.1709-c0-0-5
Degree 22
Conductor 30403040
Sign 0.995−0.0980i0.995 - 0.0980i
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  + (−0.0980 + 0.995i)2-s + (−0.485 + 1.17i)3-s + (−0.980 − 0.195i)4-s + (0.923 − 0.382i)5-s + (−1.11 − 0.598i)6-s + (0.290 − 0.956i)8-s + (−0.431 − 0.431i)9-s + (0.290 + 0.956i)10-s + (−0.636 − 1.53i)11-s + (0.704 − 1.05i)12-s + (−1.62 − 0.674i)13-s + 1.26i·15-s + (0.923 + 0.382i)16-s + (0.471 − 0.386i)18-s + (−0.923 − 0.382i)19-s + (−0.980 + 0.195i)20-s + ⋯
L(s)  = 1  + (−0.0980 + 0.995i)2-s + (−0.485 + 1.17i)3-s + (−0.980 − 0.195i)4-s + (0.923 − 0.382i)5-s + (−1.11 − 0.598i)6-s + (0.290 − 0.956i)8-s + (−0.431 − 0.431i)9-s + (0.290 + 0.956i)10-s + (−0.636 − 1.53i)11-s + (0.704 − 1.05i)12-s + (−1.62 − 0.674i)13-s + 1.26i·15-s + (0.923 + 0.382i)16-s + (0.471 − 0.386i)18-s + (−0.923 − 0.382i)19-s + (−0.980 + 0.195i)20-s + ⋯

Functional equation

Λ(s)=(3040s/2 ΓC(s) L(s)=((0.995−0.0980i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3040s/2 ΓC(s) L(s)=((0.995−0.0980i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30403040    =    25⋅5⋅192^{5} \cdot 5 \cdot 19
Sign: 0.995−0.0980i0.995 - 0.0980i
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), 0.995−0.0980i)(2,\ 3040,\ (\ :0),\ 0.995 - 0.0980i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.66139975050.6613997505
L(12)L(\frac12) ≈\approx 0.66139975050.6613997505
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+(0.0980−0.995i)T 1 + (0.0980 - 0.995i)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+(0.485−1.17i)T+(−0.707−0.707i)T2 1 + (0.485 - 1.17i)T + (-0.707 - 0.707i)T^{2}
7 1+iT2 1 + iT^{2}
11 1+(0.636+1.53i)T+(−0.707+0.707i)T2 1 + (0.636 + 1.53i)T + (-0.707 + 0.707i)T^{2}
13 1+(1.62+0.674i)T+(0.707+0.707i)T2 1 + (1.62 + 0.674i)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+(−1.83+0.761i)T+(0.707−0.707i)T2 1 + (-1.83 + 0.761i)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+(0.591+1.42i)T+(−0.707+0.707i)T2 1 + (0.591 + 1.42i)T + (-0.707 + 0.707i)T^{2}
59 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
61 1+(0.425−1.02i)T+(−0.707−0.707i)T2 1 + (0.425 - 1.02i)T + (-0.707 - 0.707i)T^{2}
67 1+(−0.732+1.76i)T+(−0.707−0.707i)T2 1 + (-0.732 + 1.76i)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 1−0.942T+T2 1 - 0.942T + 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

−8.977908305022199799240216839760, −8.219996180772111226129960218544, −7.47915756374086927150631445084, −6.32145726085141737324755166528, −5.80035890819315700977255259421, −5.05085562878261081525302158977, −4.75672765625438178658496183796, −3.62106719001609837940393254714, −2.45403260185718554689589925744, −0.44407513299358183793494515776, 1.43943600640463496749605405665, 2.23311602857593084641120276743, 2.64100705752942285895055988211, 4.32978046257333496282144650576, 4.90790196765971972943704790465, 5.88683945869376716748112407727, 6.69764653200904700338107956086, 7.42322255852609855744946757266, 7.945869708573167685691703437633, 9.253040933227427902877789536448

Graph of the ZZ-function along the critical line