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Properties

Label 2-91-1.1-c1-0-2
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s − 1.41·3-s + 4.41·5-s − 2.00·6-s + 7-s − 2.82·8-s − 0.999·9-s + 6.24·10-s − 4.24·11-s − 13-s + 1.41·14-s − 6.24·15-s − 4.00·16-s − 1.41·17-s − 1.41·18-s + 1.24·19-s − 1.41·21-s − 6·22-s − 0.171·23-s + 4·24-s + 14.4·25-s − 1.41·26-s + 5.65·27-s + 5.82·29-s − 8.82·30-s + ⋯
L(s)  = 1  + 1.00·2-s − 0.816·3-s + 1.97·5-s − 0.816·6-s + 0.377·7-s − 0.999·8-s − 0.333·9-s + 1.97·10-s − 1.27·11-s − 0.277·13-s + 0.377·14-s − 1.61·15-s − 1.00·16-s − 0.342·17-s − 0.333·18-s + 0.285·19-s − 0.308·21-s − 1.27·22-s − 0.0357·23-s + 0.816·24-s + 2.89·25-s − 0.277·26-s + 1.08·27-s + 1.08·29-s − 1.61·30-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 1.3356598921.335659892
L(12)L(\frac12) ≈\approx 1.3356598921.335659892
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 1−1.41T+2T2 1 - 1.41T + 2T^{2}
3 1+1.41T+3T2 1 + 1.41T + 3T^{2}
5 1−4.41T+5T2 1 - 4.41T + 5T^{2}
11 1+4.24T+11T2 1 + 4.24T + 11T^{2}
17 1+1.41T+17T2 1 + 1.41T + 17T^{2}
19 1−1.24T+19T2 1 - 1.24T + 19T^{2}
23 1+0.171T+23T2 1 + 0.171T + 23T^{2}
29 1−5.82T+29T2 1 - 5.82T + 29T^{2}
31 1+5.24T+31T2 1 + 5.24T + 31T^{2}
37 1+6.24T+37T2 1 + 6.24T + 37T^{2}
41 1−3.17T+41T2 1 - 3.17T + 41T^{2}
43 1+5T+43T2 1 + 5T + 43T^{2}
47 1−4.41T+47T2 1 - 4.41T + 47T^{2}
53 1+5.82T+53T2 1 + 5.82T + 53T^{2}
59 1−11.6T+59T2 1 - 11.6T + 59T^{2}
61 1−6T+61T2 1 - 6T + 61T^{2}
67 1−2.48T+67T2 1 - 2.48T + 67T^{2}
71 1−1.07T+71T2 1 - 1.07T + 71T^{2}
73 1+0.757T+73T2 1 + 0.757T + 73T^{2}
79 1+1.48T+79T2 1 + 1.48T + 79T^{2}
83 1−4.75T+83T2 1 - 4.75T + 83T^{2}
89 1−4.41T+89T2 1 - 4.41T + 89T^{2}
97 1+13.7T+97T2 1 + 13.7T + 97T^{2}
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   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

−13.91530704403553248137339920165, −13.18598525856379489294894562606, −12.31039705786425851174123193269, −10.92173925275991688173885509546, −9.956676790942336342764905946639, −8.720090509496169016660368964555, −6.55216153418865536823210553119, −5.46067764345373140398153217209, −5.09174108420603531238146202946, −2.59168801763164316544158957285, 2.59168801763164316544158957285, 5.09174108420603531238146202946, 5.46067764345373140398153217209, 6.55216153418865536823210553119, 8.720090509496169016660368964555, 9.956676790942336342764905946639, 10.92173925275991688173885509546, 12.31039705786425851174123193269, 13.18598525856379489294894562606, 13.91530704403553248137339920165

Graph of the ZZ-function along the critical line