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Properties

Label 2-4140-1.1-c1-0-17
Degree 22
Conductor 41404140
Sign 11
Analytic cond. 33.058033.0580
Root an. cond. 5.749615.74961
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 − 0.113·7-s + 3.39·11-s + 6.27·13-s + 3.61·17-s − 1.39·19-s + 23-s + 25-s − 6.38·29-s + 9.45·31-s − 0.113·35-s − 7.78·37-s + 11.0·41-s + 1.55·43-s + 1.70·47-s − 6.98·49-s − 8.71·53-s + 3.39·55-s − 8.95·59-s − 1.83·61-s + 6.27·65-s + 9.21·67-s − 6.82·71-s − 2.27·73-s − 0.387·77-s + 11.7·79-s + 1.16·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.0430·7-s + 1.02·11-s + 1.74·13-s + 0.877·17-s − 0.321·19-s + 0.208·23-s + 0.200·25-s − 1.18·29-s + 1.69·31-s − 0.0192·35-s − 1.28·37-s + 1.72·41-s + 0.237·43-s + 0.248·47-s − 0.998·49-s − 1.19·53-s + 0.458·55-s − 1.16·59-s − 0.234·61-s + 0.778·65-s + 1.12·67-s − 0.810·71-s − 0.266·73-s − 0.0441·77-s + 1.32·79-s + 0.127·83-s + ⋯

Functional equation

Λ(s)=(4140s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4140s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 41404140    =    22⋅32⋅5⋅232^{2} \cdot 3^{2} \cdot 5 \cdot 23
Sign: 11
Analytic conductor: 33.058033.0580
Root analytic conductor: 5.749615.74961
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4140, ( :1/2), 1)(2,\ 4140,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.5989313042.598931304
L(12)L(\frac12) ≈\approx 2.5989313042.598931304
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
23 1−T 1 - T
good7 1+0.113T+7T2 1 + 0.113T + 7T^{2}
11 1−3.39T+11T2 1 - 3.39T + 11T^{2}
13 1−6.27T+13T2 1 - 6.27T + 13T^{2}
17 1−3.61T+17T2 1 - 3.61T + 17T^{2}
19 1+1.39T+19T2 1 + 1.39T + 19T^{2}
29 1+6.38T+29T2 1 + 6.38T + 29T^{2}
31 1−9.45T+31T2 1 - 9.45T + 31T^{2}
37 1+7.78T+37T2 1 + 7.78T + 37T^{2}
41 1−11.0T+41T2 1 - 11.0T + 41T^{2}
43 1−1.55T+43T2 1 - 1.55T + 43T^{2}
47 1−1.70T+47T2 1 - 1.70T + 47T^{2}
53 1+8.71T+53T2 1 + 8.71T + 53T^{2}
59 1+8.95T+59T2 1 + 8.95T + 59T^{2}
61 1+1.83T+61T2 1 + 1.83T + 61T^{2}
67 1−9.21T+67T2 1 - 9.21T + 67T^{2}
71 1+6.82T+71T2 1 + 6.82T + 71T^{2}
73 1+2.27T+73T2 1 + 2.27T + 73T^{2}
79 1−11.7T+79T2 1 - 11.7T + 79T^{2}
83 1−1.16T+83T2 1 - 1.16T + 83T^{2}
89 1+1.77T+89T2 1 + 1.77T + 89T^{2}
97 1+0.131T+97T2 1 + 0.131T + 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

−8.440784537172181310575540129938, −7.79726519500570465529454551082, −6.77275550014955943701613523834, −6.18996144560893057689984751559, −5.67052356797601660698026353245, −4.58658192230373695524631545996, −3.76673764769669862155933461542, −3.08200078225406413264561959210, −1.76655044520155341346497847415, −1.00936089847603733380098820065, 1.00936089847603733380098820065, 1.76655044520155341346497847415, 3.08200078225406413264561959210, 3.76673764769669862155933461542, 4.58658192230373695524631545996, 5.67052356797601660698026353245, 6.18996144560893057689984751559, 6.77275550014955943701613523834, 7.79726519500570465529454551082, 8.440784537172181310575540129938

Graph of the ZZ-function along the critical line