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Properties

Label 2-69e2-1.1-c1-0-117
Degree 22
Conductor 47614761
Sign −1-1
Analytic cond. 38.016738.0167
Root an. cond. 6.165776.16577
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  − 2.20·2-s + 2.85·4-s − 1.23·5-s + 2.37·7-s − 1.88·8-s + 2.72·10-s − 1.91·11-s + 0.196·13-s − 5.22·14-s − 1.55·16-s − 1.55·17-s + 7.98·19-s − 3.53·20-s + 4.21·22-s − 3.47·25-s − 0.432·26-s + 6.77·28-s − 4.97·29-s + 1.78·31-s + 7.20·32-s + 3.43·34-s − 2.93·35-s − 3.88·37-s − 17.5·38-s + 2.33·40-s − 0.426·41-s − 4.45·43-s + ⋯
L(s)  = 1  − 1.55·2-s + 1.42·4-s − 0.552·5-s + 0.896·7-s − 0.666·8-s + 0.861·10-s − 0.576·11-s + 0.0544·13-s − 1.39·14-s − 0.388·16-s − 0.378·17-s + 1.83·19-s − 0.789·20-s + 0.898·22-s − 0.694·25-s − 0.0849·26-s + 1.28·28-s − 0.924·29-s + 0.320·31-s + 1.27·32-s + 0.589·34-s − 0.495·35-s − 0.638·37-s − 2.85·38-s + 0.368·40-s − 0.0666·41-s − 0.679·43-s + ⋯

Functional equation

Λ(s)=(4761s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4761s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 47614761    =    32⋅2323^{2} \cdot 23^{2}
Sign: −1-1
Analytic conductor: 38.016738.0167
Root analytic conductor: 6.165776.16577
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4761, ( :1/2), −1)(2,\ 4761,\ (\ :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)
bad3 1 1
23 1 1
good2 1+2.20T+2T2 1 + 2.20T + 2T^{2}
5 1+1.23T+5T2 1 + 1.23T + 5T^{2}
7 1−2.37T+7T2 1 - 2.37T + 7T^{2}
11 1+1.91T+11T2 1 + 1.91T + 11T^{2}
13 1−0.196T+13T2 1 - 0.196T + 13T^{2}
17 1+1.55T+17T2 1 + 1.55T + 17T^{2}
19 1−7.98T+19T2 1 - 7.98T + 19T^{2}
29 1+4.97T+29T2 1 + 4.97T + 29T^{2}
31 1−1.78T+31T2 1 - 1.78T + 31T^{2}
37 1+3.88T+37T2 1 + 3.88T + 37T^{2}
41 1+0.426T+41T2 1 + 0.426T + 41T^{2}
43 1+4.45T+43T2 1 + 4.45T + 43T^{2}
47 1−2.58T+47T2 1 - 2.58T + 47T^{2}
53 1−9.81T+53T2 1 - 9.81T + 53T^{2}
59 1−7.21T+59T2 1 - 7.21T + 59T^{2}
61 1+7.42T+61T2 1 + 7.42T + 61T^{2}
67 1+7.26T+67T2 1 + 7.26T + 67T^{2}
71 1−0.730T+71T2 1 - 0.730T + 71T^{2}
73 1+6.44T+73T2 1 + 6.44T + 73T^{2}
79 1−5.67T+79T2 1 - 5.67T + 79T^{2}
83 1+12.8T+83T2 1 + 12.8T + 83T^{2}
89 1+13.9T+89T2 1 + 13.9T + 89T^{2}
97 1+4.32T+97T2 1 + 4.32T + 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

−8.035275905461468700626322388252, −7.41570246388649369936448697018, −7.04642689800992445033568446665, −5.79563803140454523010349636505, −5.06108756205422178254252271498, −4.15053440019130459077874013523, −3.07489851055581454424072342067, −2.02002368131735351826448776789, −1.17586547274793329740610562426, 0, 1.17586547274793329740610562426, 2.02002368131735351826448776789, 3.07489851055581454424072342067, 4.15053440019130459077874013523, 5.06108756205422178254252271498, 5.79563803140454523010349636505, 7.04642689800992445033568446665, 7.41570246388649369936448697018, 8.035275905461468700626322388252

Graph of the ZZ-function along the critical line