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Properties

Label 2-3040-760.99-c0-0-1
Degree 22
Conductor 30403040
Sign −0.0389+0.999i-0.0389 + 0.999i
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  + (0.766 − 0.642i)5-s + (0.766 − 1.32i)7-s + (−0.939 + 0.342i)9-s + (−0.939 − 1.62i)11-s + (−0.173 + 0.984i)13-s + (0.939 + 0.342i)19-s + (−0.266 − 0.223i)23-s + (0.173 − 0.984i)25-s + (−0.266 − 1.50i)35-s + 0.347·37-s + (−0.326 − 1.85i)41-s + (−0.5 + 0.866i)45-s + (−0.939 + 0.342i)47-s + (−0.673 − 1.16i)49-s + (1.17 + 0.984i)53-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)5-s + (0.766 − 1.32i)7-s + (−0.939 + 0.342i)9-s + (−0.939 − 1.62i)11-s + (−0.173 + 0.984i)13-s + (0.939 + 0.342i)19-s + (−0.266 − 0.223i)23-s + (0.173 − 0.984i)25-s + (−0.266 − 1.50i)35-s + 0.347·37-s + (−0.326 − 1.85i)41-s + (−0.5 + 0.866i)45-s + (−0.939 + 0.342i)47-s + (−0.673 − 1.16i)49-s + (1.17 + 0.984i)53-s + ⋯

Functional equation

Λ(s)=(3040s/2 ΓC(s) L(s)=((−0.0389+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0389 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3040s/2 ΓC(s) L(s)=((−0.0389+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0389 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30403040    =    25⋅5⋅192^{5} \cdot 5 \cdot 19
Sign: −0.0389+0.999i-0.0389 + 0.999i
Analytic conductor: 1.517151.51715
Root analytic conductor: 1.231721.23172
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3040(1999,⋅)\chi_{3040} (1999, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3040, ( :0), −0.0389+0.999i)(2,\ 3040,\ (\ :0),\ -0.0389 + 0.999i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2835763151.283576315
L(12)L(\frac12) ≈\approx 1.2835763151.283576315
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 1
5 1+(−0.766+0.642i)T 1 + (-0.766 + 0.642i)T
19 1+(−0.939−0.342i)T 1 + (-0.939 - 0.342i)T
good3 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
7 1+(−0.766+1.32i)T+(−0.5−0.866i)T2 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2}
11 1+(0.939+1.62i)T+(−0.5+0.866i)T2 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2}
13 1+(0.173−0.984i)T+(−0.939−0.342i)T2 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2}
17 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
23 1+(0.266+0.223i)T+(0.173+0.984i)T2 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2}
29 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1−0.347T+T2 1 - 0.347T + T^{2}
41 1+(0.326+1.85i)T+(−0.939+0.342i)T2 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2}
43 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
47 1+(0.939−0.342i)T+(0.766−0.642i)T2 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2}
53 1+(−1.17−0.984i)T+(0.173+0.984i)T2 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2}
59 1+(0.939+0.342i)T+(0.766+0.642i)T2 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2}
61 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
67 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
71 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
73 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
79 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
83 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
89 1+(−0.0603+0.342i)T+(−0.939−0.342i)T2 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2}
97 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)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

−8.634274994326864181555531677541, −8.040372957764707799912439155570, −7.38301311434810973425863949403, −6.28525436236880085429473393517, −5.55051530904007454597745996651, −4.99292076794807458579569145435, −4.05763911789926881279356050319, −3.03468110327645499546955445172, −1.93927786235634076111140753869, −0.77383324126801386941563652194, 1.79108073477511873239768111000, 2.59857289482275635214955875936, 3.11933602333262775517039295237, 4.79494576040320461209077361086, 5.33881415191275298543205370536, 5.85457488962895464714455699010, 6.81334860056388656915593088441, 7.72005411528345613503314620570, 8.246816976926065541630586078055, 9.197674659148863189845754087911

Graph of the ZZ-function along the critical line