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Properties

Label 2-3960-1.1-c1-0-16
Degree 22
Conductor 39603960
Sign 11
Analytic cond. 31.620731.6207
Root an. cond. 5.623235.62323
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  + 5-s + 1.29·7-s + 11-s − 5.01·13-s + 3.29·17-s + 5.71·19-s − 5.71·23-s + 25-s + 2·29-s + 3.71·31-s + 1.29·35-s + 2·37-s + 2·41-s − 4.41·43-s + 9.71·47-s − 5.31·49-s + 4.59·53-s + 55-s − 3.71·59-s + 15.1·61-s − 5.01·65-s − 1.40·67-s + 16.0·71-s − 5.89·73-s + 1.29·77-s − 14.0·79-s + 7.52·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.491·7-s + 0.301·11-s − 1.38·13-s + 0.800·17-s + 1.31·19-s − 1.19·23-s + 0.200·25-s + 0.371·29-s + 0.666·31-s + 0.219·35-s + 0.328·37-s + 0.312·41-s − 0.672·43-s + 1.41·47-s − 0.758·49-s + 0.631·53-s + 0.134·55-s − 0.483·59-s + 1.93·61-s − 0.621·65-s − 0.171·67-s + 1.90·71-s − 0.690·73-s + 0.148·77-s − 1.57·79-s + 0.825·83-s + ⋯

Functional equation

Λ(s)=(3960s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3960s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39603960    =    23⋅32⋅5⋅112^{3} \cdot 3^{2} \cdot 5 \cdot 11
Sign: 11
Analytic conductor: 31.620731.6207
Root analytic conductor: 5.623235.62323
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3960, ( :1/2), 1)(2,\ 3960,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2315043342.231504334
L(12)L(\frac12) ≈\approx 2.2315043342.231504334
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 1
3 1 1
5 1−T 1 - T
11 1−T 1 - T
good7 1−1.29T+7T2 1 - 1.29T + 7T^{2}
13 1+5.01T+13T2 1 + 5.01T + 13T^{2}
17 1−3.29T+17T2 1 - 3.29T + 17T^{2}
19 1−5.71T+19T2 1 - 5.71T + 19T^{2}
23 1+5.71T+23T2 1 + 5.71T + 23T^{2}
29 1−2T+29T2 1 - 2T + 29T^{2}
31 1−3.71T+31T2 1 - 3.71T + 31T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1+4.41T+43T2 1 + 4.41T + 43T^{2}
47 1−9.71T+47T2 1 - 9.71T + 47T^{2}
53 1−4.59T+53T2 1 - 4.59T + 53T^{2}
59 1+3.71T+59T2 1 + 3.71T + 59T^{2}
61 1−15.1T+61T2 1 - 15.1T + 61T^{2}
67 1+1.40T+67T2 1 + 1.40T + 67T^{2}
71 1−16.0T+71T2 1 - 16.0T + 71T^{2}
73 1+5.89T+73T2 1 + 5.89T + 73T^{2}
79 1+14.0T+79T2 1 + 14.0T + 79T^{2}
83 1−7.52T+83T2 1 - 7.52T + 83T^{2}
89 1−16.5T+89T2 1 - 16.5T + 89T^{2}
97 1+6.82T+97T2 1 + 6.82T + 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

−8.359562572714099559207118510751, −7.70061909149733815088946027050, −7.11836978528448130776419983453, −6.18491206156468954017834131759, −5.39938245943619282282685189875, −4.83934062230264106406006756947, −3.87685095601942122209303665181, −2.84992029636150203322445866774, −2.01105840795360244733132285099, −0.880197024412158657214608316527, 0.880197024412158657214608316527, 2.01105840795360244733132285099, 2.84992029636150203322445866774, 3.87685095601942122209303665181, 4.83934062230264106406006756947, 5.39938245943619282282685189875, 6.18491206156468954017834131759, 7.11836978528448130776419983453, 7.70061909149733815088946027050, 8.359562572714099559207118510751

Graph of the ZZ-function along the critical line