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Properties

Label 2-3040-3040.1709-c0-0-6
Degree 22
Conductor 30403040
Sign −0.995+0.0980i-0.995 + 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+0.0980i-0.995 + 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 1.6388830801.638883080
L(12)L(\frac12) ≈\approx 1.6388830801.638883080
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

−8.742537601331100671832347364587, −8.233379943051855607107003182674, −6.97301145483287163186587506923, −6.10577288440349341980078570933, −5.63113177249924713286143902754, −4.51154903351324080162395158526, −3.43925647816447540417854639399, −2.62273278990330296652853729842, −1.74677338652528304268438416038, −1.00961141699652097689882050759, 1.85199848089373443947763673347, 3.18504112448888641683276024999, 3.87271361960550323291041956039, 4.75299912178049922290038530063, 5.39011406517105520144951906072, 6.21992341250932874928130780967, 6.90497701184722944253413711183, 7.83479409911977476370429828418, 8.617668365071651658573204656460, 9.178901576075466242790592578352

Graph of the ZZ-function along the critical line