↑

Properties

Label 2-2888-152.29-c0-0-1
Degree 22
Conductor 28882888
Sign −0.486+0.873i-0.486 + 0.873i
Analytic cond. 1.441291.44129
Root an. cond. 1.200541.20054
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  + (−0.173 − 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)6-s + (0.5 − 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.499 + 0.866i)12-s + (0.766 + 0.642i)13-s + (−0.939 − 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + (−0.173 − 0.984i)21-s + (0.939 − 0.342i)23-s + (0.939 + 0.342i)24-s + (0.766 + 0.642i)25-s + (0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)6-s + (0.5 − 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.499 + 0.866i)12-s + (0.766 + 0.642i)13-s + (−0.939 − 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + (−0.173 − 0.984i)21-s + (0.939 − 0.342i)23-s + (0.939 + 0.342i)24-s + (0.766 + 0.642i)25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

Λ(s)=(2888s/2 ΓC(s) L(s)=((−0.486+0.873i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2888s/2 ΓC(s) L(s)=((−0.486+0.873i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 28882888    =    23⋅1922^{3} \cdot 19^{2}
Sign: −0.486+0.873i-0.486 + 0.873i
Analytic conductor: 1.441291.44129
Root analytic conductor: 1.200541.20054
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2888(333,⋅)\chi_{2888} (333, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2888, ( :0), −0.486+0.873i)(2,\ 2888,\ (\ :0),\ -0.486 + 0.873i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4643532181.464353218
L(12)L(\frac12) ≈\approx 1.4643532181.464353218
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+(0.173+0.984i)T 1 + (0.173 + 0.984i)T
19 1 1
good3 1+(−0.766+0.642i)T+(0.173−0.984i)T2 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2}
5 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
7 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
13 1+(−0.766−0.642i)T+(0.173+0.984i)T2 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2}
17 1+(0.173+0.984i)T+(−0.939+0.342i)T2 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2}
23 1+(−0.939+0.342i)T+(0.766−0.642i)T2 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2}
29 1+(−0.173+0.984i)T+(−0.939−0.342i)T2 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1+2T+T2 1 + 2T + T^{2}
41 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
43 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
47 1+(−0.347+1.96i)T+(−0.939−0.342i)T2 1 + (-0.347 + 1.96i)T + (-0.939 - 0.342i)T^{2}
53 1+(0.939−0.342i)T+(0.766−0.642i)T2 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2}
59 1+(−0.173−0.984i)T+(−0.939+0.342i)T2 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2}
61 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
67 1+(−0.173+0.984i)T+(−0.939−0.342i)T2 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2}
71 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
73 1+(0.766−0.642i)T+(0.173−0.984i)T2 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2}
79 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
83 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
89 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
97 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)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

−8.790313952760150193080725258882, −8.137465274611785809562582949402, −7.30987085699735184612785784076, −6.83663449704211842075884705688, −5.31555142184851558962619040473, −4.60737062821258431640247222085, −3.69305731388831612042068609434, −2.87267701004437449148950403320, −1.92355779555376609970812347106, −1.07024134356112830537930207254, 1.44297436389906666318456100720, 2.96246869729841012150783096854, 3.70344844579570696876026231640, 4.67363137297238249845688515887, 5.37636818290611009874821690400, 6.17090498530175976737027423460, 6.91957598483210618002447179928, 7.989347371110957873463749916554, 8.599408087440597694476867418251, 8.827693263426732243548797903977

Graph of the ZZ-function along the critical line