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Properties

Label 2-2368-1.1-c1-0-41
Degree 22
Conductor 23682368
Sign −1-1
Analytic cond. 18.908518.9085
Root an. cond. 4.348394.34839
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  − 1.93·3-s + 2.93·5-s − 4.68·7-s + 0.745·9-s + 0.762·11-s − 1.76·13-s − 5.68·15-s + 3.36·17-s + 7.36·19-s + 9.06·21-s − 3.25·23-s + 3.61·25-s + 4.36·27-s + 3.25·29-s − 3.06·31-s − 1.47·33-s − 13.7·35-s + 37-s + 3.41·39-s − 7.42·41-s − 12.2·43-s + 2.18·45-s − 0.302·47-s + 14.9·49-s − 6.50·51-s − 5.53·53-s + 2.23·55-s + ⋯
L(s)  = 1  − 1.11·3-s + 1.31·5-s − 1.76·7-s + 0.248·9-s + 0.229·11-s − 0.488·13-s − 1.46·15-s + 0.815·17-s + 1.68·19-s + 1.97·21-s − 0.678·23-s + 0.723·25-s + 0.839·27-s + 0.604·29-s − 0.550·31-s − 0.256·33-s − 2.32·35-s + 0.164·37-s + 0.546·39-s − 1.15·41-s − 1.86·43-s + 0.326·45-s − 0.0440·47-s + 2.13·49-s − 0.911·51-s − 0.760·53-s + 0.301·55-s + ⋯

Functional equation

Λ(s)=(2368s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2368s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23682368    =    26⋅372^{6} \cdot 37
Sign: −1-1
Analytic conductor: 18.908518.9085
Root analytic conductor: 4.348394.34839
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2368, ( :1/2), −1)(2,\ 2368,\ (\ :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
37 1−T 1 - T
good3 1+1.93T+3T2 1 + 1.93T + 3T^{2}
5 1−2.93T+5T2 1 - 2.93T + 5T^{2}
7 1+4.68T+7T2 1 + 4.68T + 7T^{2}
11 1−0.762T+11T2 1 - 0.762T + 11T^{2}
13 1+1.76T+13T2 1 + 1.76T + 13T^{2}
17 1−3.36T+17T2 1 - 3.36T + 17T^{2}
19 1−7.36T+19T2 1 - 7.36T + 19T^{2}
23 1+3.25T+23T2 1 + 3.25T + 23T^{2}
29 1−3.25T+29T2 1 - 3.25T + 29T^{2}
31 1+3.06T+31T2 1 + 3.06T + 31T^{2}
41 1+7.42T+41T2 1 + 7.42T + 41T^{2}
43 1+12.2T+43T2 1 + 12.2T + 43T^{2}
47 1+0.302T+47T2 1 + 0.302T + 47T^{2}
53 1+5.53T+53T2 1 + 5.53T + 53T^{2}
59 1−10.2T+59T2 1 - 10.2T + 59T^{2}
61 1+12.2T+61T2 1 + 12.2T + 61T^{2}
67 1+13.1T+67T2 1 + 13.1T + 67T^{2}
71 1+0.173T+71T2 1 + 0.173T + 71T^{2}
73 1+1.23T+73T2 1 + 1.23T + 73T^{2}
79 1+4.61T+79T2 1 + 4.61T + 79T^{2}
83 1−3.53T+83T2 1 - 3.53T + 83T^{2}
89 1−15.7T+89T2 1 - 15.7T + 89T^{2}
97 1+16.1T+97T2 1 + 16.1T + 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

−8.902386820949328232085049668707, −7.58386915294965718612748495956, −6.65695905187835746597161811998, −6.24303256703266197267961835147, −5.55734847331483286213249208269, −5.02027819319793879495335637939, −3.50115944702038849552515004838, −2.79363622965920750775064069811, −1.37093618209581543756882175601, 0, 1.37093618209581543756882175601, 2.79363622965920750775064069811, 3.50115944702038849552515004838, 5.02027819319793879495335637939, 5.55734847331483286213249208269, 6.24303256703266197267961835147, 6.65695905187835746597161811998, 7.58386915294965718612748495956, 8.902386820949328232085049668707

Graph of the ZZ-function along the critical line