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Properties

Label 2-1682-1.1-c1-0-49
Degree 22
Conductor 16821682
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2.24·3-s + 4-s + 1.55·5-s + 2.24·6-s + 3.49·7-s + 8-s + 2.04·9-s + 1.55·10-s − 3.74·11-s + 2.24·12-s − 3.85·13-s + 3.49·14-s + 3.49·15-s + 16-s + 4.93·17-s + 2.04·18-s + 6.15·19-s + 1.55·20-s + 7.85·21-s − 3.74·22-s − 5.76·23-s + 2.24·24-s − 2.58·25-s − 3.85·26-s − 2.13·27-s + 3.49·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.29·3-s + 0.5·4-s + 0.695·5-s + 0.917·6-s + 1.32·7-s + 0.353·8-s + 0.682·9-s + 0.491·10-s − 1.12·11-s + 0.648·12-s − 1.06·13-s + 0.933·14-s + 0.902·15-s + 0.250·16-s + 1.19·17-s + 0.482·18-s + 1.41·19-s + 0.347·20-s + 1.71·21-s − 0.797·22-s − 1.20·23-s + 0.458·24-s − 0.516·25-s − 0.755·26-s − 0.411·27-s + 0.660·28-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: 11
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: 00
Selberg data: (2, 1682, ( :1/2), 1)(2,\ 1682,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.9231528574.923152857
L(12)L(\frac12) ≈\approx 4.9231528574.923152857
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 1−2.24T+3T2 1 - 2.24T + 3T^{2}
5 1−1.55T+5T2 1 - 1.55T + 5T^{2}
7 1−3.49T+7T2 1 - 3.49T + 7T^{2}
11 1+3.74T+11T2 1 + 3.74T + 11T^{2}
13 1+3.85T+13T2 1 + 3.85T + 13T^{2}
17 1−4.93T+17T2 1 - 4.93T + 17T^{2}
19 1−6.15T+19T2 1 - 6.15T + 19T^{2}
23 1+5.76T+23T2 1 + 5.76T + 23T^{2}
31 1+2.51T+31T2 1 + 2.51T + 31T^{2}
37 1+6.78T+37T2 1 + 6.78T + 37T^{2}
41 1+0.0271T+41T2 1 + 0.0271T + 41T^{2}
43 1−11.7T+43T2 1 - 11.7T + 43T^{2}
47 1−0.554T+47T2 1 - 0.554T + 47T^{2}
53 1−3.25T+53T2 1 - 3.25T + 53T^{2}
59 1+3.06T+59T2 1 + 3.06T + 59T^{2}
61 1+0.131T+61T2 1 + 0.131T + 61T^{2}
67 1+11T+67T2 1 + 11T + 67T^{2}
71 1+3.62T+71T2 1 + 3.62T + 71T^{2}
73 1−6.62T+73T2 1 - 6.62T + 73T^{2}
79 1+1.50T+79T2 1 + 1.50T + 79T^{2}
83 1+9.96T+83T2 1 + 9.96T + 83T^{2}
89 1−5.39T+89T2 1 - 5.39T + 89T^{2}
97 1+4.31T+97T2 1 + 4.31T + 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

−9.436253820321982142678948813552, −8.343671427229291284966682989418, −7.62569224962550377882452005503, −7.43959448582500253927460332165, −5.65161625202525762540718433678, −5.37680658228399008509620060644, −4.32359227633140756332062589567, −3.21629652050587479329959357881, −2.41494598642847298886400091075, −1.66981753705387663269958363527, 1.66981753705387663269958363527, 2.41494598642847298886400091075, 3.21629652050587479329959357881, 4.32359227633140756332062589567, 5.37680658228399008509620060644, 5.65161625202525762540718433678, 7.43959448582500253927460332165, 7.62569224962550377882452005503, 8.343671427229291284966682989418, 9.436253820321982142678948813552

Graph of the ZZ-function along the critical line