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Properties

Label 2-91e2-1.1-c1-0-202
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  + 0.456·2-s − 2.79·3-s − 1.79·4-s − 0.456·5-s − 1.27·6-s − 1.73·8-s + 4.79·9-s − 0.208·10-s − 3.92·11-s + 5·12-s + 1.27·15-s + 2.79·16-s + 3·17-s + 2.18·18-s − 1.37·19-s + 0.818·20-s − 1.79·22-s + 1.58·23-s + 4.83·24-s − 4.79·25-s − 4.99·27-s − 6.79·29-s + 0.582·30-s − 8.66·31-s + 4.73·32-s + 10.9·33-s + 1.37·34-s + ⋯
L(s)  = 1  + 0.323·2-s − 1.61·3-s − 0.895·4-s − 0.204·5-s − 0.520·6-s − 0.612·8-s + 1.59·9-s − 0.0660·10-s − 1.18·11-s + 1.44·12-s + 0.329·15-s + 0.697·16-s + 0.727·17-s + 0.515·18-s − 0.314·19-s + 0.182·20-s − 0.381·22-s + 0.329·23-s + 0.986·24-s − 0.958·25-s − 0.962·27-s − 1.26·29-s + 0.106·30-s − 1.55·31-s + 0.837·32-s + 1.90·33-s + 0.235·34-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 1−0.456T+2T2 1 - 0.456T + 2T^{2}
3 1+2.79T+3T2 1 + 2.79T + 3T^{2}
5 1+0.456T+5T2 1 + 0.456T + 5T^{2}
11 1+3.92T+11T2 1 + 3.92T + 11T^{2}
17 1−3T+17T2 1 - 3T + 17T^{2}
19 1+1.37T+19T2 1 + 1.37T + 19T^{2}
23 1−1.58T+23T2 1 - 1.58T + 23T^{2}
29 1+6.79T+29T2 1 + 6.79T + 29T^{2}
31 1+8.66T+31T2 1 + 8.66T + 31T^{2}
37 1−6.92T+37T2 1 - 6.92T + 37T^{2}
41 1−7.84T+41T2 1 - 7.84T + 41T^{2}
43 1+9.37T+43T2 1 + 9.37T + 43T^{2}
47 1+9.57T+47T2 1 + 9.57T + 47T^{2}
53 1−6.16T+53T2 1 - 6.16T + 53T^{2}
59 1−12.3T+59T2 1 - 12.3T + 59T^{2}
61 1−14.7T+61T2 1 - 14.7T + 61T^{2}
67 1−4.47T+67T2 1 - 4.47T + 67T^{2}
71 1−4.37T+71T2 1 - 4.37T + 71T^{2}
73 1+3.46T+73T2 1 + 3.46T + 73T^{2}
79 1+6T+79T2 1 + 6T + 79T^{2}
83 1−7.02T+83T2 1 - 7.02T + 83T^{2}
89 1−16.1T+89T2 1 - 16.1T + 89T^{2}
97 1−7.28T+97T2 1 - 7.28T + 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

−7.46165394824936461986637080256, −6.56629245457661557270617377360, −5.71570223720788005835521735337, −5.43408824262829632141378971710, −4.90976747342396851652808757254, −4.04806462246680120676345033995, −3.43463228160997762014783205089, −2.12184621440783039473598579230, −0.791226229635397120914887217480, 0, 0.791226229635397120914887217480, 2.12184621440783039473598579230, 3.43463228160997762014783205089, 4.04806462246680120676345033995, 4.90976747342396851652808757254, 5.43408824262829632141378971710, 5.71570223720788005835521735337, 6.56629245457661557270617377360, 7.46165394824936461986637080256

Graph of the ZZ-function along the critical line