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Properties

Label 2-91e2-1.1-c1-0-222
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  − 2.38·2-s − 2.75·3-s + 3.70·4-s + 0.982·5-s + 6.57·6-s − 4.06·8-s + 4.57·9-s − 2.34·10-s − 0.587·11-s − 10.1·12-s − 2.70·15-s + 2.30·16-s − 6.45·17-s − 10.9·18-s − 3.82·19-s + 3.63·20-s + 1.40·22-s + 8.26·23-s + 11.1·24-s − 4.03·25-s − 4.32·27-s − 3.96·29-s + 6.45·30-s − 2.98·31-s + 2.62·32-s + 1.61·33-s + 15.4·34-s + ⋯
L(s)  = 1  − 1.68·2-s − 1.58·3-s + 1.85·4-s + 0.439·5-s + 2.68·6-s − 1.43·8-s + 1.52·9-s − 0.741·10-s − 0.177·11-s − 2.94·12-s − 0.697·15-s + 0.576·16-s − 1.56·17-s − 2.57·18-s − 0.877·19-s + 0.813·20-s + 0.299·22-s + 1.72·23-s + 2.28·24-s − 0.807·25-s − 0.831·27-s − 0.735·29-s + 1.17·30-s − 0.536·31-s + 0.464·32-s + 0.281·33-s + 2.64·34-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+2.38T+2T2 1 + 2.38T + 2T^{2}
3 1+2.75T+3T2 1 + 2.75T + 3T^{2}
5 1−0.982T+5T2 1 - 0.982T + 5T^{2}
11 1+0.587T+11T2 1 + 0.587T + 11T^{2}
17 1+6.45T+17T2 1 + 6.45T + 17T^{2}
19 1+3.82T+19T2 1 + 3.82T + 19T^{2}
23 1−8.26T+23T2 1 - 8.26T + 23T^{2}
29 1+3.96T+29T2 1 + 3.96T + 29T^{2}
31 1+2.98T+31T2 1 + 2.98T + 31T^{2}
37 1−1.75T+37T2 1 - 1.75T + 37T^{2}
41 1−3.67T+41T2 1 - 3.67T + 41T^{2}
43 1−6.38T+43T2 1 - 6.38T + 43T^{2}
47 1+4.34T+47T2 1 + 4.34T + 47T^{2}
53 1−0.425T+53T2 1 - 0.425T + 53T^{2}
59 1−6.00T+59T2 1 - 6.00T + 59T^{2}
61 1−2.20T+61T2 1 - 2.20T + 61T^{2}
67 1−7.01T+67T2 1 - 7.01T + 67T^{2}
71 1−3.60T+71T2 1 - 3.60T + 71T^{2}
73 1−4.93T+73T2 1 - 4.93T + 73T^{2}
79 1−2.78T+79T2 1 - 2.78T + 79T^{2}
83 1+2.86T+83T2 1 + 2.86T + 83T^{2}
89 1+2.09T+89T2 1 + 2.09T + 89T^{2}
97 1−7.69T+97T2 1 - 7.69T + 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

−7.35406491711751018747600687146, −6.80625573833046745289519899946, −6.35736710814792257046606104729, −5.62579527414048991257808862727, −4.88302420129346247591177472128, −4.05086090158787778710198752280, −2.53879277040556516444517951049, −1.83304595444883949530609874929, −0.830673765686925700893816157909, 0, 0.830673765686925700893816157909, 1.83304595444883949530609874929, 2.53879277040556516444517951049, 4.05086090158787778710198752280, 4.88302420129346247591177472128, 5.62579527414048991257808862727, 6.35736710814792257046606104729, 6.80625573833046745289519899946, 7.35406491711751018747600687146

Graph of the ZZ-function along the critical line