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Properties

Label 2-91e2-1.1-c1-0-364
Degree 22
Conductor 82818281
Sign −1-1
Analytic cond. 66.124166.1241
Root an. cond. 8.131678.13167
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.656·2-s − 0.204·3-s − 1.56·4-s + 1.35·5-s − 0.134·6-s − 2.34·8-s − 2.95·9-s + 0.892·10-s + 1.90·11-s + 0.321·12-s − 0.278·15-s + 1.60·16-s + 3.56·17-s − 1.94·18-s − 0.985·19-s − 2.13·20-s + 1.25·22-s + 1.69·23-s + 0.479·24-s − 3.15·25-s + 1.21·27-s + 6.54·29-s − 0.182·30-s − 7.69·31-s + 5.73·32-s − 0.390·33-s + 2.34·34-s + ⋯
L(s)  = 1  + 0.463·2-s − 0.118·3-s − 0.784·4-s + 0.608·5-s − 0.0548·6-s − 0.828·8-s − 0.986·9-s + 0.282·10-s + 0.574·11-s + 0.0927·12-s − 0.0718·15-s + 0.400·16-s + 0.864·17-s − 0.457·18-s − 0.226·19-s − 0.477·20-s + 0.266·22-s + 0.353·23-s + 0.0978·24-s − 0.630·25-s + 0.234·27-s + 1.21·29-s − 0.0333·30-s − 1.38·31-s + 1.01·32-s − 0.0679·33-s + 0.401·34-s + ⋯

Functional equation

Λ(s)=(8281s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(8281s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 82818281    =    72⋅1327^{2} \cdot 13^{2}
Sign: −1-1
Analytic conductor: 66.124166.1241
Root analytic conductor: 8.131678.13167
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 8281, ( :1/2), −1)(2,\ 8281,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 1
13 1 1
good2 1−0.656T+2T2 1 - 0.656T + 2T^{2}
3 1+0.204T+3T2 1 + 0.204T + 3T^{2}
5 1−1.35T+5T2 1 - 1.35T + 5T^{2}
11 1−1.90T+11T2 1 - 1.90T + 11T^{2}
17 1−3.56T+17T2 1 - 3.56T + 17T^{2}
19 1+0.985T+19T2 1 + 0.985T + 19T^{2}
23 1−1.69T+23T2 1 - 1.69T + 23T^{2}
29 1−6.54T+29T2 1 - 6.54T + 29T^{2}
31 1+7.69T+31T2 1 + 7.69T + 31T^{2}
37 1−2.02T+37T2 1 - 2.02T + 37T^{2}
41 1+9.88T+41T2 1 + 9.88T + 41T^{2}
43 1+3.16T+43T2 1 + 3.16T + 43T^{2}
47 1+7.76T+47T2 1 + 7.76T + 47T^{2}
53 1−0.354T+53T2 1 - 0.354T + 53T^{2}
59 1+2.16T+59T2 1 + 2.16T + 59T^{2}
61 1−12.2T+61T2 1 - 12.2T + 61T^{2}
67 1+11.3T+67T2 1 + 11.3T + 67T^{2}
71 1−9.05T+71T2 1 - 9.05T + 71T^{2}
73 1−7.13T+73T2 1 - 7.13T + 73T^{2}
79 1+5.39T+79T2 1 + 5.39T + 79T^{2}
83 1−2.03T+83T2 1 - 2.03T + 83T^{2}
89 1−6.89T+89T2 1 - 6.89T + 89T^{2}
97 1−14.6T+97T2 1 - 14.6T + 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

−7.47471045594710939824721421469, −6.40692748606181296530646447718, −6.08317807152053092865562182046, −5.19065346780208609813190192519, −4.94660266588747611222586140547, −3.74938636099151667818091477826, −3.33537730490894149019626877718, −2.33577796029993186202790757595, −1.21399960076514160510021024523, 0, 1.21399960076514160510021024523, 2.33577796029993186202790757595, 3.33537730490894149019626877718, 3.74938636099151667818091477826, 4.94660266588747611222586140547, 5.19065346780208609813190192519, 6.08317807152053092865562182046, 6.40692748606181296530646447718, 7.47471045594710939824721421469

Graph of the ZZ-function along the critical line