↑

Properties

Label 2-3750-1.1-c1-0-43
Degree 22
Conductor 37503750
Sign −1-1
Analytic cond. 29.943929.9439
Root an. cond. 5.472105.47210
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  − 2-s − 3-s + 4-s + 6-s − 3.36·7-s − 8-s + 9-s + 4.03·11-s − 12-s + 2.91·13-s + 3.36·14-s + 16-s + 1.20·17-s − 18-s − 0.591·19-s + 3.36·21-s − 4.03·22-s − 8.53·23-s + 24-s − 2.91·26-s − 27-s − 3.36·28-s − 6.30·29-s − 0.0361·31-s − 32-s − 4.03·33-s − 1.20·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.27·7-s − 0.353·8-s + 0.333·9-s + 1.21·11-s − 0.288·12-s + 0.807·13-s + 0.899·14-s + 0.250·16-s + 0.293·17-s − 0.235·18-s − 0.135·19-s + 0.734·21-s − 0.860·22-s − 1.78·23-s + 0.204·24-s − 0.571·26-s − 0.192·27-s − 0.635·28-s − 1.17·29-s − 0.00649·31-s − 0.176·32-s − 0.702·33-s − 0.207·34-s + ⋯

Functional equation

Λ(s)=(3750s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3750s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37503750    =    2⋅3⋅542 \cdot 3 \cdot 5^{4}
Sign: −1-1
Analytic conductor: 29.943929.9439
Root analytic conductor: 5.472105.47210
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3750, ( :1/2), −1)(2,\ 3750,\ (\ :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+T 1 + T
3 1+T 1 + T
5 1 1
good7 1+3.36T+7T2 1 + 3.36T + 7T^{2}
11 1−4.03T+11T2 1 - 4.03T + 11T^{2}
13 1−2.91T+13T2 1 - 2.91T + 13T^{2}
17 1−1.20T+17T2 1 - 1.20T + 17T^{2}
19 1+0.591T+19T2 1 + 0.591T + 19T^{2}
23 1+8.53T+23T2 1 + 8.53T + 23T^{2}
29 1+6.30T+29T2 1 + 6.30T + 29T^{2}
31 1+0.0361T+31T2 1 + 0.0361T + 31T^{2}
37 1+10.4T+37T2 1 + 10.4T + 37T^{2}
41 1−2.19T+41T2 1 - 2.19T + 41T^{2}
43 1−9.45T+43T2 1 - 9.45T + 43T^{2}
47 1−5.43T+47T2 1 - 5.43T + 47T^{2}
53 1−6.55T+53T2 1 - 6.55T + 53T^{2}
59 1−0.178T+59T2 1 - 0.178T + 59T^{2}
61 1−0.646T+61T2 1 - 0.646T + 61T^{2}
67 1−9.50T+67T2 1 - 9.50T + 67T^{2}
71 1−8.74T+71T2 1 - 8.74T + 71T^{2}
73 1+16.0T+73T2 1 + 16.0T + 73T^{2}
79 1−11.3T+79T2 1 - 11.3T + 79T^{2}
83 1−16.4T+83T2 1 - 16.4T + 83T^{2}
89 1+1.40T+89T2 1 + 1.40T + 89T^{2}
97 1+8.63T+97T2 1 + 8.63T + 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

−8.200383318400314560923082427535, −7.31514377178040429033481166042, −6.60599992001677399692022379319, −6.11326745571300895563046672396, −5.48318221541314529454002954556, −3.90166622641528482648839876743, −3.70132000143096434653106430809, −2.28407058842610371251835161401, −1.19380171109081369838334441094, 0, 1.19380171109081369838334441094, 2.28407058842610371251835161401, 3.70132000143096434653106430809, 3.90166622641528482648839876743, 5.48318221541314529454002954556, 6.11326745571300895563046672396, 6.60599992001677399692022379319, 7.31514377178040429033481166042, 8.200383318400314560923082427535

Graph of the ZZ-function along the critical line