↑

Properties

Label 2-91-1.1-c1-0-0
Degree 22
Conductor 9191
Sign 11
Analytic cond. 0.7266380.726638
Root an. cond. 0.8524310.852431
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  − 1.81·2-s − 3.10·3-s + 1.28·4-s + 2.81·5-s + 5.62·6-s − 7-s + 1.28·8-s + 6.62·9-s − 5.10·10-s + 3.10·11-s − 3.99·12-s + 13-s + 1.81·14-s − 8.72·15-s − 4.91·16-s − 0.524·17-s − 12.0·18-s + 0.813·19-s + 3.62·20-s + 3.10·21-s − 5.62·22-s + 7.33·23-s − 4.00·24-s + 2.91·25-s − 1.81·26-s − 11.2·27-s − 1.28·28-s + ⋯
L(s)  = 1  − 1.28·2-s − 1.79·3-s + 0.644·4-s + 1.25·5-s + 2.29·6-s − 0.377·7-s + 0.455·8-s + 2.20·9-s − 1.61·10-s + 0.935·11-s − 1.15·12-s + 0.277·13-s + 0.484·14-s − 2.25·15-s − 1.22·16-s − 0.127·17-s − 2.83·18-s + 0.186·19-s + 0.811·20-s + 0.677·21-s − 1.19·22-s + 1.53·23-s − 0.816·24-s + 0.583·25-s − 0.355·26-s − 2.16·27-s − 0.243·28-s + ⋯

Functional equation

Λ(s)=(91s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(91s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9191    =    7⋅137 \cdot 13
Sign: 11
Analytic conductor: 0.7266380.726638
Root analytic conductor: 0.8524310.852431
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 91, ( :1/2), 1)(2,\ 91,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.38810122280.3881012228
L(12)L(\frac12) ≈\approx 0.38810122280.3881012228
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)
bad7 1+T 1 + T
13 1−T 1 - T
good2 1+1.81T+2T2 1 + 1.81T + 2T^{2}
3 1+3.10T+3T2 1 + 3.10T + 3T^{2}
5 1−2.81T+5T2 1 - 2.81T + 5T^{2}
11 1−3.10T+11T2 1 - 3.10T + 11T^{2}
17 1+0.524T+17T2 1 + 0.524T + 17T^{2}
19 1−0.813T+19T2 1 - 0.813T + 19T^{2}
23 1−7.33T+23T2 1 - 7.33T + 23T^{2}
29 1−8.28T+29T2 1 - 8.28T + 29T^{2}
31 1−1.39T+31T2 1 - 1.39T + 31T^{2}
37 1+6.15T+37T2 1 + 6.15T + 37T^{2}
41 1+4.20T+41T2 1 + 4.20T + 41T^{2}
43 1−6.75T+43T2 1 - 6.75T + 43T^{2}
47 1+5.97T+47T2 1 + 5.97T + 47T^{2}
53 1+2.49T+53T2 1 + 2.49T + 53T^{2}
59 1+4.47T+59T2 1 + 4.47T + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1−10.0T+67T2 1 - 10.0T + 67T^{2}
71 1+8.72T+71T2 1 + 8.72T + 71T^{2}
73 1+2.34T+73T2 1 + 2.34T + 73T^{2}
79 1+13.5T+79T2 1 + 13.5T + 79T^{2}
83 1−16.4T+83T2 1 - 16.4T + 83T^{2}
89 1+10.6T+89T2 1 + 10.6T + 89T^{2}
97 1+1.18T+97T2 1 + 1.18T + 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

−13.88777830242513739692539289281, −12.80973583034402151499416733291, −11.59086863743711495577045499367, −10.60099224119543009400133905705, −9.895812578457702131950705188115, −8.948365161903600243579050520670, −6.96372729183181824285675067939, −6.22024670683491927018005029540, −4.88149030753689217866982759788, −1.24925895028719379559500876429, 1.24925895028719379559500876429, 4.88149030753689217866982759788, 6.22024670683491927018005029540, 6.96372729183181824285675067939, 8.948365161903600243579050520670, 9.895812578457702131950705188115, 10.60099224119543009400133905705, 11.59086863743711495577045499367, 12.80973583034402151499416733291, 13.88777830242513739692539289281

Graph of the ZZ-function along the critical line