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Properties

Label 2-3744-1.1-c1-0-50
Degree 22
Conductor 37443744
Sign −1-1
Analytic cond. 29.895929.8959
Root an. cond. 5.467725.46772
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  + 2.96·5-s − 3.35·7-s − 1.61·11-s + 13-s − 2·17-s + 3.35·19-s − 6.70·23-s + 3.77·25-s − 2·29-s − 6.57·31-s − 9.92·35-s + 7.92·37-s − 6.96·41-s + 0.775·43-s + 2.38·47-s + 4.22·49-s − 11.9·53-s − 4.77·55-s + 0.312·59-s + 14.6·61-s + 2.96·65-s − 8.12·67-s − 4.31·71-s + 0.0752·73-s + 5.40·77-s − 12·79-s − 8.31·83-s + ⋯
L(s)  = 1  + 1.32·5-s − 1.26·7-s − 0.486·11-s + 0.277·13-s − 0.485·17-s + 0.768·19-s − 1.39·23-s + 0.755·25-s − 0.371·29-s − 1.18·31-s − 1.67·35-s + 1.30·37-s − 1.08·41-s + 0.118·43-s + 0.348·47-s + 0.603·49-s − 1.63·53-s − 0.643·55-s + 0.0407·59-s + 1.87·61-s + 0.367·65-s − 0.992·67-s − 0.511·71-s + 0.00880·73-s + 0.615·77-s − 1.35·79-s − 0.912·83-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 37443744    =    25⋅32⋅132^{5} \cdot 3^{2} \cdot 13
Sign: −1-1
Analytic conductor: 29.895929.8959
Root analytic conductor: 5.467725.46772
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3744, ( :1/2), −1)(2,\ 3744,\ (\ :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
13 1−T 1 - T
good5 1−2.96T+5T2 1 - 2.96T + 5T^{2}
7 1+3.35T+7T2 1 + 3.35T + 7T^{2}
11 1+1.61T+11T2 1 + 1.61T + 11T^{2}
17 1+2T+17T2 1 + 2T + 17T^{2}
19 1−3.35T+19T2 1 - 3.35T + 19T^{2}
23 1+6.70T+23T2 1 + 6.70T + 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1+6.57T+31T2 1 + 6.57T + 31T^{2}
37 1−7.92T+37T2 1 - 7.92T + 37T^{2}
41 1+6.96T+41T2 1 + 6.96T + 41T^{2}
43 1−0.775T+43T2 1 - 0.775T + 43T^{2}
47 1−2.38T+47T2 1 - 2.38T + 47T^{2}
53 1+11.9T+53T2 1 + 11.9T + 53T^{2}
59 1−0.312T+59T2 1 - 0.312T + 59T^{2}
61 1−14.6T+61T2 1 - 14.6T + 61T^{2}
67 1+8.12T+67T2 1 + 8.12T + 67T^{2}
71 1+4.31T+71T2 1 + 4.31T + 71T^{2}
73 1−0.0752T+73T2 1 - 0.0752T + 73T^{2}
79 1+12T+79T2 1 + 12T + 79T^{2}
83 1+8.31T+83T2 1 + 8.31T + 83T^{2}
89 1+8.88T+89T2 1 + 8.88T + 89T^{2}
97 1+7.92T+97T2 1 + 7.92T + 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

−8.208513110453880992616491427738, −7.27219446418248418344265826066, −6.53493475651358488646551380391, −5.86237492087646943619293221464, −5.46020096559540566522851589011, −4.26169233818429002857627996313, −3.30326384920402681131239776789, −2.49631119810651316349746927676, −1.58679715639157800024865947374, 0, 1.58679715639157800024865947374, 2.49631119810651316349746927676, 3.30326384920402681131239776789, 4.26169233818429002857627996313, 5.46020096559540566522851589011, 5.86237492087646943619293221464, 6.53493475651358488646551380391, 7.27219446418248418344265826066, 8.208513110453880992616491427738

Graph of the ZZ-function along the critical line