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Properties

Label 2-1638-1.1-c1-0-7
Degree 22
Conductor 16381638
Sign 11
Analytic cond. 13.079413.0794
Root an. cond. 3.616553.61655
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 + 4-s − 2.37·5-s − 7-s + 8-s − 2.37·10-s + 2.37·11-s − 13-s − 14-s + 16-s + 4.37·17-s + 1.62·19-s − 2.37·20-s + 2.37·22-s + 3.62·23-s + 0.627·25-s − 26-s − 28-s + 6.37·29-s − 4.74·31-s + 32-s + 4.37·34-s + 2.37·35-s − 4.37·37-s + 1.62·38-s − 2.37·40-s + 8.74·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.06·5-s − 0.377·7-s + 0.353·8-s − 0.750·10-s + 0.715·11-s − 0.277·13-s − 0.267·14-s + 0.250·16-s + 1.06·17-s + 0.373·19-s − 0.530·20-s + 0.505·22-s + 0.756·23-s + 0.125·25-s − 0.196·26-s − 0.188·28-s + 1.18·29-s − 0.852·31-s + 0.176·32-s + 0.749·34-s + 0.400·35-s − 0.718·37-s + 0.264·38-s − 0.375·40-s + 1.36·41-s + ⋯

Functional equation

Λ(s)=(1638s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1638s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16381638    =    2⋅32⋅7⋅132 \cdot 3^{2} \cdot 7 \cdot 13
Sign: 11
Analytic conductor: 13.079413.0794
Root analytic conductor: 3.616553.61655
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1638, ( :1/2), 1)(2,\ 1638,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2603852462.260385246
L(12)L(\frac12) ≈\approx 2.2603852462.260385246
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
3 1 1
7 1+T 1 + T
13 1+T 1 + T
good5 1+2.37T+5T2 1 + 2.37T + 5T^{2}
11 1−2.37T+11T2 1 - 2.37T + 11T^{2}
17 1−4.37T+17T2 1 - 4.37T + 17T^{2}
19 1−1.62T+19T2 1 - 1.62T + 19T^{2}
23 1−3.62T+23T2 1 - 3.62T + 23T^{2}
29 1−6.37T+29T2 1 - 6.37T + 29T^{2}
31 1+4.74T+31T2 1 + 4.74T + 31T^{2}
37 1+4.37T+37T2 1 + 4.37T + 37T^{2}
41 1−8.74T+41T2 1 - 8.74T + 41T^{2}
43 1−11.1T+43T2 1 - 11.1T + 43T^{2}
47 1−1.25T+47T2 1 - 1.25T + 47T^{2}
53 1−8.74T+53T2 1 - 8.74T + 53T^{2}
59 1−2T+59T2 1 - 2T + 59T^{2}
61 1−5.11T+61T2 1 - 5.11T + 61T^{2}
67 1−9.48T+67T2 1 - 9.48T + 67T^{2}
71 1+4.74T+71T2 1 + 4.74T + 71T^{2}
73 1+8.37T+73T2 1 + 8.37T + 73T^{2}
79 1−4.74T+79T2 1 - 4.74T + 79T^{2}
83 1+6T+83T2 1 + 6T + 83T^{2}
89 1+3.25T+89T2 1 + 3.25T + 89T^{2}
97 1−7.48T+97T2 1 - 7.48T + 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

−9.373857463487479114668721141890, −8.511085555264429188428361004414, −7.51167601164705766351562169422, −7.10947571004955242301607573170, −6.06076001080333560473021100537, −5.22538541035983250319292524494, −4.18261892703617307841785715116, −3.59714856281968611883874867388, −2.64434025492289001675448653364, −0.979531254790084051525936085842, 0.979531254790084051525936085842, 2.64434025492289001675448653364, 3.59714856281968611883874867388, 4.18261892703617307841785715116, 5.22538541035983250319292524494, 6.06076001080333560473021100537, 7.10947571004955242301607573170, 7.51167601164705766351562169422, 8.511085555264429188428361004414, 9.373857463487479114668721141890

Graph of the ZZ-function along the critical line