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Properties

Label 2-1682-1.1-c1-0-25
Degree 22
Conductor 16821682
Sign −1-1
Analytic cond. 13.430813.4308
Root an. cond. 3.664813.66481
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-s − 3.41·3-s + 4-s + 1.61·5-s + 3.41·6-s − 1.52·7-s − 8-s + 8.67·9-s − 1.61·10-s − 1.89·11-s − 3.41·12-s − 3.18·13-s + 1.52·14-s − 5.52·15-s + 16-s + 1.69·17-s − 8.67·18-s + 3.67·19-s + 1.61·20-s + 5.19·21-s + 1.89·22-s − 2.69·23-s + 3.41·24-s − 2.39·25-s + 3.18·26-s − 19.4·27-s − 1.52·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.97·3-s + 0.5·4-s + 0.722·5-s + 1.39·6-s − 0.574·7-s − 0.353·8-s + 2.89·9-s − 0.510·10-s − 0.571·11-s − 0.986·12-s − 0.883·13-s + 0.406·14-s − 1.42·15-s + 0.250·16-s + 0.412·17-s − 2.04·18-s + 0.844·19-s + 0.361·20-s + 1.13·21-s + 0.404·22-s − 0.561·23-s + 0.697·24-s − 0.478·25-s + 0.624·26-s − 3.73·27-s − 0.287·28-s + ⋯

Functional equation

Λ(s)=(1682s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1682s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16821682    =    2⋅2922 \cdot 29^{2}
Sign: −1-1
Analytic conductor: 13.430813.4308
Root analytic conductor: 3.664813.66481
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1682, ( :1/2), −1)(2,\ 1682,\ (\ :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)
bad2 1+T 1 + T
29 1 1
good3 1+3.41T+3T2 1 + 3.41T + 3T^{2}
5 1−1.61T+5T2 1 - 1.61T + 5T^{2}
7 1+1.52T+7T2 1 + 1.52T + 7T^{2}
11 1+1.89T+11T2 1 + 1.89T + 11T^{2}
13 1+3.18T+13T2 1 + 3.18T + 13T^{2}
17 1−1.69T+17T2 1 - 1.69T + 17T^{2}
19 1−3.67T+19T2 1 - 3.67T + 19T^{2}
23 1+2.69T+23T2 1 + 2.69T + 23T^{2}
31 1−1.78T+31T2 1 - 1.78T + 31T^{2}
37 1−7.07T+37T2 1 - 7.07T + 37T^{2}
41 1−5.03T+41T2 1 - 5.03T + 41T^{2}
43 1−1.89T+43T2 1 - 1.89T + 43T^{2}
47 1+2.57T+47T2 1 + 2.57T + 47T^{2}
53 1−8.24T+53T2 1 - 8.24T + 53T^{2}
59 1−2.95T+59T2 1 - 2.95T + 59T^{2}
61 1−7.74T+61T2 1 - 7.74T + 61T^{2}
67 1−2.47T+67T2 1 - 2.47T + 67T^{2}
71 1−6.06T+71T2 1 - 6.06T + 71T^{2}
73 1+3.74T+73T2 1 + 3.74T + 73T^{2}
79 1+9.22T+79T2 1 + 9.22T + 79T^{2}
83 1+5.52T+83T2 1 + 5.52T + 83T^{2}
89 1+15.9T+89T2 1 + 15.9T + 89T^{2}
97 1+14.9T+97T2 1 + 14.9T + 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

−9.491280252919751660318713113705, −7.923906410911647185758370628821, −7.20761301954186033087634011418, −6.47697304193846975504108272103, −5.70829067831697462114606752740, −5.28056377802281550576302211956, −4.12394005569430865376691996099, −2.50357072987484932210345331178, −1.19884447357088848587887105758, 0, 1.19884447357088848587887105758, 2.50357072987484932210345331178, 4.12394005569430865376691996099, 5.28056377802281550576302211956, 5.70829067831697462114606752740, 6.47697304193846975504108272103, 7.20761301954186033087634011418, 7.923906410911647185758370628821, 9.491280252919751660318713113705

Graph of the ZZ-function along the critical line