↑

Properties

Label 2-3744-1.1-c1-0-46
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.561·5-s + 0.561·7-s + 2·11-s − 13-s + 0.561·17-s − 6·19-s − 4.68·25-s + 8.24·29-s − 7.12·31-s − 0.315·35-s − 9.68·37-s − 7.12·41-s + 8.80·43-s + 1.68·47-s − 6.68·49-s + 4.87·53-s − 1.12·55-s − 6·59-s + 13.3·61-s + 0.561·65-s − 6·67-s − 1.68·71-s + 10·73-s + 1.12·77-s − 12·79-s − 17.3·83-s − 0.315·85-s + ⋯
L(s)  = 1  − 0.251·5-s + 0.212·7-s + 0.603·11-s − 0.277·13-s + 0.136·17-s − 1.37·19-s − 0.936·25-s + 1.53·29-s − 1.27·31-s − 0.0533·35-s − 1.59·37-s − 1.11·41-s + 1.34·43-s + 0.245·47-s − 0.954·49-s + 0.669·53-s − 0.151·55-s − 0.781·59-s + 1.71·61-s + 0.0696·65-s − 0.733·67-s − 0.199·71-s + 1.17·73-s + 0.127·77-s − 1.35·79-s − 1.90·83-s − 0.0342·85-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+0.561T+5T2 1 + 0.561T + 5T^{2}
7 1−0.561T+7T2 1 - 0.561T + 7T^{2}
11 1−2T+11T2 1 - 2T + 11T^{2}
17 1−0.561T+17T2 1 - 0.561T + 17T^{2}
19 1+6T+19T2 1 + 6T + 19T^{2}
23 1+23T2 1 + 23T^{2}
29 1−8.24T+29T2 1 - 8.24T + 29T^{2}
31 1+7.12T+31T2 1 + 7.12T + 31T^{2}
37 1+9.68T+37T2 1 + 9.68T + 37T^{2}
41 1+7.12T+41T2 1 + 7.12T + 41T^{2}
43 1−8.80T+43T2 1 - 8.80T + 43T^{2}
47 1−1.68T+47T2 1 - 1.68T + 47T^{2}
53 1−4.87T+53T2 1 - 4.87T + 53T^{2}
59 1+6T+59T2 1 + 6T + 59T^{2}
61 1−13.3T+61T2 1 - 13.3T + 61T^{2}
67 1+6T+67T2 1 + 6T + 67T^{2}
71 1+1.68T+71T2 1 + 1.68T + 71T^{2}
73 1−10T+73T2 1 - 10T + 73T^{2}
79 1+12T+79T2 1 + 12T + 79T^{2}
83 1+17.3T+83T2 1 + 17.3T + 83T^{2}
89 1−8.24T+89T2 1 - 8.24T + 89T^{2}
97 1+6T+97T2 1 + 6T + 97T^{2}
show more
show less
   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

−8.310080589934033374812375342089, −7.33713268528036076349461049532, −6.74069179733242319110287892029, −5.94728239294354965613847389707, −5.08379743035954149505590828294, −4.23642977810300404332541306626, −3.58569493556401252252580430744, −2.43470077497720609517368412384, −1.49264770548250293594637876598, 0, 1.49264770548250293594637876598, 2.43470077497720609517368412384, 3.58569493556401252252580430744, 4.23642977810300404332541306626, 5.08379743035954149505590828294, 5.94728239294354965613847389707, 6.74069179733242319110287892029, 7.33713268528036076349461049532, 8.310080589934033374812375342089

Graph of the ZZ-function along the critical line