↑

Properties

Label 2-1254-1.1-c1-0-9
Degree 22
Conductor 12541254
Sign 11
Analytic cond. 10.013210.0132
Root an. cond. 3.164373.16437
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 3.23·5-s + 6-s + 2·7-s + 8-s + 9-s − 3.23·10-s − 11-s + 12-s + 0.763·13-s + 2·14-s − 3.23·15-s + 16-s + 4.47·17-s + 18-s + 19-s − 3.23·20-s + 2·21-s − 22-s + 7.70·23-s + 24-s + 5.47·25-s + 0.763·26-s + 27-s + 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.44·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 0.333·9-s − 1.02·10-s − 0.301·11-s + 0.288·12-s + 0.211·13-s + 0.534·14-s − 0.835·15-s + 0.250·16-s + 1.08·17-s + 0.235·18-s + 0.229·19-s − 0.723·20-s + 0.436·21-s − 0.213·22-s + 1.60·23-s + 0.204·24-s + 1.09·25-s + 0.149·26-s + 0.192·27-s + 0.377·28-s + ⋯

Functional equation

Λ(s)=(1254s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1254s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12541254    =    2⋅3⋅11⋅192 \cdot 3 \cdot 11 \cdot 19
Sign: 11
Analytic conductor: 10.013210.0132
Root analytic conductor: 3.164373.16437
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1254, ( :1/2), 1)(2,\ 1254,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.8120513252.812051325
L(12)L(\frac12) ≈\approx 2.8120513252.812051325
L(32)L(\frac{3}{2}) 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−T 1 - T
3 1−T 1 - T
11 1+T 1 + T
19 1−T 1 - T
good5 1+3.23T+5T2 1 + 3.23T + 5T^{2}
7 1−2T+7T2 1 - 2T + 7T^{2}
13 1−0.763T+13T2 1 - 0.763T + 13T^{2}
17 1−4.47T+17T2 1 - 4.47T + 17T^{2}
23 1−7.70T+23T2 1 - 7.70T + 23T^{2}
29 1−2T+29T2 1 - 2T + 29T^{2}
31 1−7.23T+31T2 1 - 7.23T + 31T^{2}
37 1−5.23T+37T2 1 - 5.23T + 37T^{2}
41 1+3.52T+41T2 1 + 3.52T + 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1+7.70T+47T2 1 + 7.70T + 47T^{2}
53 1+8.94T+53T2 1 + 8.94T + 53T^{2}
59 1+2.47T+59T2 1 + 2.47T + 59T^{2}
61 1−4T+61T2 1 - 4T + 61T^{2}
67 1−4T+67T2 1 - 4T + 67T^{2}
71 1+12.4T+71T2 1 + 12.4T + 71T^{2}
73 1−10.9T+73T2 1 - 10.9T + 73T^{2}
79 1+9.23T+79T2 1 + 9.23T + 79T^{2}
83 1+4T+83T2 1 + 4T + 83T^{2}
89 1−10.9T+89T2 1 - 10.9T + 89T^{2}
97 1−10.9T+97T2 1 - 10.9T + 97T^{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

−9.728869631079103147539874899268, −8.540553645059729523992104427175, −7.961979839717142258750964307095, −7.44106196520399250966194683031, −6.46500892459673180295756569395, −5.09985948165748119112158820363, −4.54651272707773043293834808348, −3.51536257709699716727703330746, −2.85717303269076822508916571979, −1.20462983542259946438258617303, 1.20462983542259946438258617303, 2.85717303269076822508916571979, 3.51536257709699716727703330746, 4.54651272707773043293834808348, 5.09985948165748119112158820363, 6.46500892459673180295756569395, 7.44106196520399250966194683031, 7.961979839717142258750964307095, 8.540553645059729523992104427175, 9.728869631079103147539874899268

Graph of the ZZ-function along the critical line