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Properties

Label 2-3744-13.12-c1-0-44
Degree 22
Conductor 37443744
Sign ii
Analytic cond. 29.895929.8959
Root an. cond. 5.467725.46772
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.60i·7-s + 6.60i·11-s − 3.60·13-s − 7.21·17-s + 1.39i·19-s + 5·25-s + 7.21·29-s − 10.6i·31-s − 5.39i·47-s + 0.211·49-s + 2·53-s − 11.8i·59-s + 6·61-s − 14.6i·67-s − 15.8i·71-s + ⋯
L(s)  = 1  − 0.984i·7-s + 1.99i·11-s − 1.00·13-s − 1.74·17-s + 0.319i·19-s + 25-s + 1.33·29-s − 1.90i·31-s − 0.786i·47-s + 0.0301·49-s + 0.274·53-s − 1.53i·59-s + 0.768·61-s − 1.78i·67-s − 1.87i·71-s + ⋯

Functional equation

Λ(s)=(3744s/2 ΓC(s) L(s)=(i Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3744s/2 ΓC(s+1/2) L(s)=(i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37443744    =    25⋅32⋅132^{5} \cdot 3^{2} \cdot 13
Sign: ii
Analytic conductor: 29.895929.8959
Root analytic conductor: 5.467725.46772
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3744(3457,⋅)\chi_{3744} (3457, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3744, ( :1/2), i)(2,\ 3744,\ (\ :1/2),\ i)

Particular Values

L(1)L(1) ≈\approx 1.0984014201.098401420
L(12)L(\frac12) ≈\approx 1.0984014201.098401420
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
13 1+3.60T 1 + 3.60T
good5 1−5T2 1 - 5T^{2}
7 1+2.60iT−7T2 1 + 2.60iT - 7T^{2}
11 1−6.60iT−11T2 1 - 6.60iT - 11T^{2}
17 1+7.21T+17T2 1 + 7.21T + 17T^{2}
19 1−1.39iT−19T2 1 - 1.39iT - 19T^{2}
23 1+23T2 1 + 23T^{2}
29 1−7.21T+29T2 1 - 7.21T + 29T^{2}
31 1+10.6iT−31T2 1 + 10.6iT - 31T^{2}
37 1−37T2 1 - 37T^{2}
41 1−41T2 1 - 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1+5.39iT−47T2 1 + 5.39iT - 47T^{2}
53 1−2T+53T2 1 - 2T + 53T^{2}
59 1+11.8iT−59T2 1 + 11.8iT - 59T^{2}
61 1−6T+61T2 1 - 6T + 61T^{2}
67 1+14.6iT−67T2 1 + 14.6iT - 67T^{2}
71 1+15.8iT−71T2 1 + 15.8iT - 71T^{2}
73 1−73T2 1 - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1+3.81iT−83T2 1 + 3.81iT - 83T^{2}
89 1−89T2 1 - 89T^{2}
97 1−97T2 1 - 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

−8.188714410022765953348898517419, −7.46687973426303549897576943307, −6.91211612798683437875753461131, −6.40338724760419898388595810205, −4.93669865433510563417066547880, −4.61034845843340959257084786086, −3.90150650028817211998177541828, −2.52576631237108973117886350179, −1.88677364854305106408275258683, −0.35327816287155092958314919044, 1.04491903159100579149462532736, 2.64351768618998015962007007946, 2.83171472885535519983873340301, 4.15532604203154359013918325815, 5.05208776354983522619073681004, 5.64847975517670532648088371738, 6.55210619196176382595558219291, 7.01177430795672550093713925931, 8.297620885046749219286082070497, 8.730348733941722924493114525274

Graph of the ZZ-function along the critical line