↑

Properties

Label 2-115-1.1-c1-0-0
Degree 22
Conductor 115115
Sign 11
Analytic cond. 0.9182790.918279
Root an. cond. 0.9582690.958269
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  − 1.69·2-s − 2.56·3-s + 0.868·4-s + 5-s + 4.33·6-s − 0.819·7-s + 1.91·8-s + 3.56·9-s − 1.69·10-s + 5.38·11-s − 2.22·12-s + 2.46·13-s + 1.38·14-s − 2.56·15-s − 4.98·16-s + 4.20·17-s − 6.03·18-s − 5.38·19-s + 0.868·20-s + 2.09·21-s − 9.12·22-s − 23-s − 4.91·24-s + 25-s − 4.17·26-s − 1.43·27-s − 0.710·28-s + ⋯
L(s)  = 1  − 1.19·2-s − 1.47·3-s + 0.434·4-s + 0.447·5-s + 1.77·6-s − 0.309·7-s + 0.677·8-s + 1.18·9-s − 0.535·10-s + 1.62·11-s − 0.641·12-s + 0.683·13-s + 0.370·14-s − 0.661·15-s − 1.24·16-s + 1.02·17-s − 1.42·18-s − 1.23·19-s + 0.194·20-s + 0.457·21-s − 1.94·22-s − 0.208·23-s − 1.00·24-s + 0.200·25-s − 0.818·26-s − 0.276·27-s − 0.134·28-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: 11
Analytic conductor: 0.9182790.918279
Root analytic conductor: 0.9582690.958269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 115, ( :1/2), 1)(2,\ 115,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.40508470230.4050847023
L(12)L(\frac12) ≈\approx 0.40508470230.4050847023
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)
bad5 1−T 1 - T
23 1+T 1 + T
good2 1+1.69T+2T2 1 + 1.69T + 2T^{2}
3 1+2.56T+3T2 1 + 2.56T + 3T^{2}
7 1+0.819T+7T2 1 + 0.819T + 7T^{2}
11 1−5.38T+11T2 1 - 5.38T + 11T^{2}
13 1−2.46T+13T2 1 - 2.46T + 13T^{2}
17 1−4.20T+17T2 1 - 4.20T + 17T^{2}
19 1+5.38T+19T2 1 + 5.38T + 19T^{2}
29 1−2.35T+29T2 1 - 2.35T + 29T^{2}
31 1−6.66T+31T2 1 - 6.66T + 31T^{2}
37 1−7.42T+37T2 1 - 7.42T + 37T^{2}
41 1−5.64T+41T2 1 - 5.64T + 41T^{2}
43 1+1.90T+43T2 1 + 1.90T + 43T^{2}
47 1+9.33T+47T2 1 + 9.33T + 47T^{2}
53 1−13.8T+53T2 1 - 13.8T + 53T^{2}
59 1−5.27T+59T2 1 - 5.27T + 59T^{2}
61 1+8.51T+61T2 1 + 8.51T + 61T^{2}
67 1−4.10T+67T2 1 - 4.10T + 67T^{2}
71 1+11.7T+71T2 1 + 11.7T + 71T^{2}
73 1+10.4T+73T2 1 + 10.4T + 73T^{2}
79 1−13.7T+79T2 1 - 13.7T + 79T^{2}
83 1−3.32T+83T2 1 - 3.32T + 83T^{2}
89 1+3.58T+89T2 1 + 3.58T + 89T^{2}
97 1+9.02T+97T2 1 + 9.02T + 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

−13.39044580534011118362598037612, −12.16639880919803416218905565180, −11.28230148328621455379942078355, −10.33482649441925781624541748506, −9.523336456815885102911488976996, −8.367300205792908111679029711403, −6.74179140659092668547956801214, −6.02785255257524048626341650016, −4.37006479462296182814920270020, −1.14760301018491783932398264453, 1.14760301018491783932398264453, 4.37006479462296182814920270020, 6.02785255257524048626341650016, 6.74179140659092668547956801214, 8.367300205792908111679029711403, 9.523336456815885102911488976996, 10.33482649441925781624541748506, 11.28230148328621455379942078355, 12.16639880919803416218905565180, 13.39044580534011118362598037612

Graph of the ZZ-function along the critical line