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Properties

Label 2-4140-1.1-c1-0-22
Degree 22
Conductor 41404140
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 3.44·7-s − 2.44·11-s + 4.44·13-s + 5.44·17-s − 1.55·19-s + 23-s + 25-s + 1.89·29-s − 7·31-s + 3.44·35-s + 6.34·37-s + 7.89·41-s − 8.89·43-s − 2.44·47-s + 4.89·49-s − 4.34·53-s + 2.44·55-s − 1.89·59-s − 5.34·61-s − 4.44·65-s + 1.44·67-s + 3·71-s + 9.34·73-s + 8.44·77-s − 4·79-s − 10.3·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.30·7-s − 0.738·11-s + 1.23·13-s + 1.32·17-s − 0.355·19-s + 0.208·23-s + 0.200·25-s + 0.352·29-s − 1.25·31-s + 0.583·35-s + 1.04·37-s + 1.23·41-s − 1.35·43-s − 0.357·47-s + 0.699·49-s − 0.597·53-s + 0.330·55-s − 0.247·59-s − 0.684·61-s − 0.551·65-s + 0.177·67-s + 0.356·71-s + 1.09·73-s + 0.962·77-s − 0.450·79-s − 1.13·83-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: −1-1
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: 11
Selberg data: (2, 4140, ( :1/2), −1)(2,\ 4140,\ (\ :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 1
3 1 1
5 1+T 1 + T
23 1−T 1 - T
good7 1+3.44T+7T2 1 + 3.44T + 7T^{2}
11 1+2.44T+11T2 1 + 2.44T + 11T^{2}
13 1−4.44T+13T2 1 - 4.44T + 13T^{2}
17 1−5.44T+17T2 1 - 5.44T + 17T^{2}
19 1+1.55T+19T2 1 + 1.55T + 19T^{2}
29 1−1.89T+29T2 1 - 1.89T + 29T^{2}
31 1+7T+31T2 1 + 7T + 31T^{2}
37 1−6.34T+37T2 1 - 6.34T + 37T^{2}
41 1−7.89T+41T2 1 - 7.89T + 41T^{2}
43 1+8.89T+43T2 1 + 8.89T + 43T^{2}
47 1+2.44T+47T2 1 + 2.44T + 47T^{2}
53 1+4.34T+53T2 1 + 4.34T + 53T^{2}
59 1+1.89T+59T2 1 + 1.89T + 59T^{2}
61 1+5.34T+61T2 1 + 5.34T + 61T^{2}
67 1−1.44T+67T2 1 - 1.44T + 67T^{2}
71 1−3T+71T2 1 - 3T + 71T^{2}
73 1−9.34T+73T2 1 - 9.34T + 73T^{2}
79 1+4T+79T2 1 + 4T + 79T^{2}
83 1+10.3T+83T2 1 + 10.3T + 83T^{2}
89 1+16.8T+89T2 1 + 16.8T + 89T^{2}
97 1+14.8T+97T2 1 + 14.8T + 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

−8.027086375797784856924669577004, −7.38669223646632197481899076236, −6.49866880258402391661377494326, −5.94248484526938516102918064744, −5.16117494249830695247356575562, −4.02835155466474128318589351458, −3.39225186097387603644406133265, −2.72600835508057881818660331510, −1.26222459211938058750437967144, 0, 1.26222459211938058750437967144, 2.72600835508057881818660331510, 3.39225186097387603644406133265, 4.02835155466474128318589351458, 5.16117494249830695247356575562, 5.94248484526938516102918064744, 6.49866880258402391661377494326, 7.38669223646632197481899076236, 8.027086375797784856924669577004

Graph of the ZZ-function along the critical line