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Properties

Label 2-1098-1.1-c1-0-0
Degree 22
Conductor 10981098
Sign 11
Analytic cond. 8.767578.76757
Root an. cond. 2.961002.96100
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  − 2-s + 4-s − 1.39·5-s − 3.18·7-s − 8-s + 1.39·10-s − 0.323·11-s + 2.32·13-s + 3.18·14-s + 16-s − 1.72·17-s − 2.86·19-s − 1.39·20-s + 0.323·22-s + 6.50·23-s − 3.04·25-s − 2.32·26-s − 3.18·28-s − 0.0643·29-s + 5.10·31-s − 32-s + 1.72·34-s + 4.45·35-s + 8.98·37-s + 2.86·38-s + 1.39·40-s + 2.65·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.625·5-s − 1.20·7-s − 0.353·8-s + 0.442·10-s − 0.0975·11-s + 0.644·13-s + 0.851·14-s + 0.250·16-s − 0.417·17-s − 0.656·19-s − 0.312·20-s + 0.0689·22-s + 1.35·23-s − 0.609·25-s − 0.455·26-s − 0.601·28-s − 0.0119·29-s + 0.917·31-s − 0.176·32-s + 0.295·34-s + 0.752·35-s + 1.47·37-s + 0.464·38-s + 0.221·40-s + 0.414·41-s + ⋯

Functional equation

Λ(s)=(1098s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1098s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10981098    =    2⋅32⋅612 \cdot 3^{2} \cdot 61
Sign: 11
Analytic conductor: 8.767578.76757
Root analytic conductor: 2.961002.96100
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1098, ( :1/2), 1)(2,\ 1098,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.78035477640.7803547764
L(12)L(\frac12) ≈\approx 0.78035477640.7803547764
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 1
61 1+T 1 + T
good5 1+1.39T+5T2 1 + 1.39T + 5T^{2}
7 1+3.18T+7T2 1 + 3.18T + 7T^{2}
11 1+0.323T+11T2 1 + 0.323T + 11T^{2}
13 1−2.32T+13T2 1 - 2.32T + 13T^{2}
17 1+1.72T+17T2 1 + 1.72T + 17T^{2}
19 1+2.86T+19T2 1 + 2.86T + 19T^{2}
23 1−6.50T+23T2 1 - 6.50T + 23T^{2}
29 1+0.0643T+29T2 1 + 0.0643T + 29T^{2}
31 1−5.10T+31T2 1 - 5.10T + 31T^{2}
37 1−8.98T+37T2 1 - 8.98T + 37T^{2}
41 1−2.65T+41T2 1 - 2.65T + 41T^{2}
43 1−11.4T+43T2 1 - 11.4T + 43T^{2}
47 1−4.79T+47T2 1 - 4.79T + 47T^{2}
53 1+12.9T+53T2 1 + 12.9T + 53T^{2}
59 1−6.60T+59T2 1 - 6.60T + 59T^{2}
67 1+4.69T+67T2 1 + 4.69T + 67T^{2}
71 1+3.41T+71T2 1 + 3.41T + 71T^{2}
73 1−13.9T+73T2 1 - 13.9T + 73T^{2}
79 1−4.19T+79T2 1 - 4.19T + 79T^{2}
83 1−16.1T+83T2 1 - 16.1T + 83T^{2}
89 1+2.51T+89T2 1 + 2.51T + 89T^{2}
97 1+6.18T+97T2 1 + 6.18T + 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

−9.651277935830380482623882154380, −9.150524933110662642054948910507, −8.248342970226689120037345636894, −7.46969977897348269598395783305, −6.55103099382117253862630666796, −5.96953127185242466684810529717, −4.48430178871476826379961281754, −3.47837216586679871804272213063, −2.50191690690808572462807579836, −0.73014257456017341271987376300, 0.73014257456017341271987376300, 2.50191690690808572462807579836, 3.47837216586679871804272213063, 4.48430178871476826379961281754, 5.96953127185242466684810529717, 6.55103099382117253862630666796, 7.46969977897348269598395783305, 8.248342970226689120037345636894, 9.150524933110662642054948910507, 9.651277935830380482623882154380

Graph of the ZZ-function along the critical line