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Properties

Label 2-693-1.1-c3-0-33
Degree 22
Conductor 693693
Sign 11
Analytic cond. 40.888340.8883
Root an. cond. 6.394396.39439
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.21·2-s − 3.08·4-s + 22.0·5-s − 7·7-s − 24.5·8-s + 48.8·10-s − 11·11-s + 2.76·13-s − 15.5·14-s − 29.7·16-s + 50.6·17-s + 56.5·19-s − 68.0·20-s − 24.3·22-s + 172.·23-s + 360.·25-s + 6.13·26-s + 21.6·28-s − 282.·29-s + 237.·31-s + 130.·32-s + 112.·34-s − 154.·35-s + 207.·37-s + 125.·38-s − 541.·40-s − 386.·41-s + ⋯
L(s)  = 1  + 0.783·2-s − 0.386·4-s + 1.97·5-s − 0.377·7-s − 1.08·8-s + 1.54·10-s − 0.301·11-s + 0.0590·13-s − 0.296·14-s − 0.464·16-s + 0.722·17-s + 0.682·19-s − 0.761·20-s − 0.236·22-s + 1.56·23-s + 2.88·25-s + 0.0462·26-s + 0.145·28-s − 1.80·29-s + 1.37·31-s + 0.721·32-s + 0.566·34-s − 0.745·35-s + 0.920·37-s + 0.534·38-s − 2.14·40-s − 1.47·41-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: 11
Analytic conductor: 40.888340.8883
Root analytic conductor: 6.394396.39439
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 693, ( :3/2), 1)(2,\ 693,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.6545764543.654576454
L(12)L(\frac12) ≈\approx 3.6545764543.654576454
L(52)L(\frac{5}{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
7 1+7T 1 + 7T
11 1+11T 1 + 11T
good2 1−2.21T+8T2 1 - 2.21T + 8T^{2}
5 1−22.0T+125T2 1 - 22.0T + 125T^{2}
13 1−2.76T+2.19e3T2 1 - 2.76T + 2.19e3T^{2}
17 1−50.6T+4.91e3T2 1 - 50.6T + 4.91e3T^{2}
19 1−56.5T+6.85e3T2 1 - 56.5T + 6.85e3T^{2}
23 1−172.T+1.21e4T2 1 - 172.T + 1.21e4T^{2}
29 1+282.T+2.43e4T2 1 + 282.T + 2.43e4T^{2}
31 1−237.T+2.97e4T2 1 - 237.T + 2.97e4T^{2}
37 1−207.T+5.06e4T2 1 - 207.T + 5.06e4T^{2}
41 1+386.T+6.89e4T2 1 + 386.T + 6.89e4T^{2}
43 1+18.3T+7.95e4T2 1 + 18.3T + 7.95e4T^{2}
47 1−309.T+1.03e5T2 1 - 309.T + 1.03e5T^{2}
53 1−480.T+1.48e5T2 1 - 480.T + 1.48e5T^{2}
59 1+114.T+2.05e5T2 1 + 114.T + 2.05e5T^{2}
61 1−109.T+2.26e5T2 1 - 109.T + 2.26e5T^{2}
67 1+567.T+3.00e5T2 1 + 567.T + 3.00e5T^{2}
71 1−780.T+3.57e5T2 1 - 780.T + 3.57e5T^{2}
73 1+605.T+3.89e5T2 1 + 605.T + 3.89e5T^{2}
79 1−686.T+4.93e5T2 1 - 686.T + 4.93e5T^{2}
83 1+316.T+5.71e5T2 1 + 316.T + 5.71e5T^{2}
89 1−1.61e3T+7.04e5T2 1 - 1.61e3T + 7.04e5T^{2}
97 1+1.23e3T+9.12e5T2 1 + 1.23e3T + 9.12e5T^{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

−9.850077558308333407642651123101, −9.425518644258138302195217914962, −8.624046254762289435025889291261, −7.13329046136621270068806198138, −6.12572047941256166886702454952, −5.50980137732487071677128003588, −4.87260921949520813706180408123, −3.37103794082095769803280396569, −2.49900577162958189467731688533, −1.04405278275181453343337043252, 1.04405278275181453343337043252, 2.49900577162958189467731688533, 3.37103794082095769803280396569, 4.87260921949520813706180408123, 5.50980137732487071677128003588, 6.12572047941256166886702454952, 7.13329046136621270068806198138, 8.624046254762289435025889291261, 9.425518644258138302195217914962, 9.850077558308333407642651123101

Graph of the ZZ-function along the critical line