↑

Properties

Label 2-115-1.1-c3-0-1
Degree 22
Conductor 115115
Sign 11
Analytic cond. 6.785216.78521
Root an. cond. 2.604842.60484
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.0457·2-s − 10.2·3-s − 7.99·4-s + 5·5-s − 0.467·6-s − 33.8·7-s − 0.732·8-s + 77.4·9-s + 0.228·10-s − 21.0·11-s + 81.7·12-s + 32.8·13-s − 1.54·14-s − 51.0·15-s + 63.9·16-s − 28.7·17-s + 3.54·18-s + 67.2·19-s − 39.9·20-s + 346.·21-s − 0.962·22-s + 23·23-s + 7.48·24-s + 25·25-s + 1.50·26-s − 515.·27-s + 270.·28-s + ⋯
L(s)  = 1  + 0.0161·2-s − 1.96·3-s − 0.999·4-s + 0.447·5-s − 0.0318·6-s − 1.82·7-s − 0.0323·8-s + 2.86·9-s + 0.00723·10-s − 0.576·11-s + 1.96·12-s + 0.701·13-s − 0.0295·14-s − 0.879·15-s + 0.999·16-s − 0.409·17-s + 0.0464·18-s + 0.812·19-s − 0.447·20-s + 3.59·21-s − 0.00932·22-s + 0.208·23-s + 0.0636·24-s + 0.200·25-s + 0.0113·26-s − 3.67·27-s + 1.82·28-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: 11
Analytic conductor: 6.785216.78521
Root analytic conductor: 2.604842.60484
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 115, ( :3/2), 1)(2,\ 115,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.40963789180.4096378918
L(12)L(\frac12) ≈\approx 0.40963789180.4096378918
L(52)L(\frac{5}{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 1−5T 1 - 5T
23 1−23T 1 - 23T
good2 1−0.0457T+8T2 1 - 0.0457T + 8T^{2}
3 1+10.2T+27T2 1 + 10.2T + 27T^{2}
7 1+33.8T+343T2 1 + 33.8T + 343T^{2}
11 1+21.0T+1.33e3T2 1 + 21.0T + 1.33e3T^{2}
13 1−32.8T+2.19e3T2 1 - 32.8T + 2.19e3T^{2}
17 1+28.7T+4.91e3T2 1 + 28.7T + 4.91e3T^{2}
19 1−67.2T+6.85e3T2 1 - 67.2T + 6.85e3T^{2}
29 1+199.T+2.43e4T2 1 + 199.T + 2.43e4T^{2}
31 1−2.47T+2.97e4T2 1 - 2.47T + 2.97e4T^{2}
37 1+173.T+5.06e4T2 1 + 173.T + 5.06e4T^{2}
41 1−200.T+6.89e4T2 1 - 200.T + 6.89e4T^{2}
43 1−198.T+7.95e4T2 1 - 198.T + 7.95e4T^{2}
47 1−33.4T+1.03e5T2 1 - 33.4T + 1.03e5T^{2}
53 1−556.T+1.48e5T2 1 - 556.T + 1.48e5T^{2}
59 1−234.T+2.05e5T2 1 - 234.T + 2.05e5T^{2}
61 1−26.2T+2.26e5T2 1 - 26.2T + 2.26e5T^{2}
67 1−190.T+3.00e5T2 1 - 190.T + 3.00e5T^{2}
71 1−745.T+3.57e5T2 1 - 745.T + 3.57e5T^{2}
73 1+742.T+3.89e5T2 1 + 742.T + 3.89e5T^{2}
79 1−715.T+4.93e5T2 1 - 715.T + 4.93e5T^{2}
83 1−766.T+5.71e5T2 1 - 766.T + 5.71e5T^{2}
89 1+683.T+7.04e5T2 1 + 683.T + 7.04e5T^{2}
97 1+1.17e3T+9.12e5T2 1 + 1.17e3T + 9.12e5T^{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

−13.01900088309646621380526718782, −12.25539238915022634107517798230, −10.88165945493859993582384400909, −10.01058547559569213935077954047, −9.279927742719726671081680563055, −7.13545668566898270368315203117, −6.02491637508595939800395644725, −5.35056184412338496762954718137, −3.84710920530781106597727684870, −0.60087811501644249758278501196, 0.60087811501644249758278501196, 3.84710920530781106597727684870, 5.35056184412338496762954718137, 6.02491637508595939800395644725, 7.13545668566898270368315203117, 9.279927742719726671081680563055, 10.01058547559569213935077954047, 10.88165945493859993582384400909, 12.25539238915022634107517798230, 13.01900088309646621380526718782

Graph of the ZZ-function along the critical line