↑

Properties

Label 2-2793-1.1-c1-0-36
Degree 22
Conductor 27932793
Sign 11
Analytic cond. 22.302222.3022
Root an. cond. 4.722524.72252
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 4-s + 2·5-s − 6-s − 3·8-s + 9-s + 2·10-s + 6.24·11-s + 12-s − 0.828·13-s − 2·15-s − 16-s + 0.828·17-s + 18-s + 19-s − 2·20-s + 6.24·22-s − 2.24·23-s + 3·24-s − 25-s − 0.828·26-s − 27-s − 1.65·29-s − 2·30-s + 0.828·31-s + 5·32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 0.5·4-s + 0.894·5-s − 0.408·6-s − 1.06·8-s + 0.333·9-s + 0.632·10-s + 1.88·11-s + 0.288·12-s − 0.229·13-s − 0.516·15-s − 0.250·16-s + 0.200·17-s + 0.235·18-s + 0.229·19-s − 0.447·20-s + 1.33·22-s − 0.467·23-s + 0.612·24-s − 0.200·25-s − 0.162·26-s − 0.192·27-s − 0.307·29-s − 0.365·30-s + 0.148·31-s + 0.883·32-s + ⋯

Functional equation

Λ(s)=(2793s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2793s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27932793    =    3⋅72⋅193 \cdot 7^{2} \cdot 19
Sign: 11
Analytic conductor: 22.302222.3022
Root analytic conductor: 4.722524.72252
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2793, ( :1/2), 1)(2,\ 2793,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2778715132.277871513
L(12)L(\frac12) ≈\approx 2.2778715132.277871513
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)
bad3 1+T 1 + T
7 1 1
19 1−T 1 - T
good2 1−T+2T2 1 - T + 2T^{2}
5 1−2T+5T2 1 - 2T + 5T^{2}
11 1−6.24T+11T2 1 - 6.24T + 11T^{2}
13 1+0.828T+13T2 1 + 0.828T + 13T^{2}
17 1−0.828T+17T2 1 - 0.828T + 17T^{2}
23 1+2.24T+23T2 1 + 2.24T + 23T^{2}
29 1+1.65T+29T2 1 + 1.65T + 29T^{2}
31 1−0.828T+31T2 1 - 0.828T + 31T^{2}
37 1−7.65T+37T2 1 - 7.65T + 37T^{2}
41 1−3.07T+41T2 1 - 3.07T + 41T^{2}
43 1−1.17T+43T2 1 - 1.17T + 43T^{2}
47 1+10.4T+47T2 1 + 10.4T + 47T^{2}
53 1−11.6T+53T2 1 - 11.6T + 53T^{2}
59 1+5.65T+59T2 1 + 5.65T + 59T^{2}
61 1+4.24T+61T2 1 + 4.24T + 61T^{2}
67 1−12.5T+67T2 1 - 12.5T + 67T^{2}
71 1−5.17T+71T2 1 - 5.17T + 71T^{2}
73 1−1.41T+73T2 1 - 1.41T + 73T^{2}
79 1−6.24T+79T2 1 - 6.24T + 79T^{2}
83 1−12T+83T2 1 - 12T + 83T^{2}
89 1+16.7T+89T2 1 + 16.7T + 89T^{2}
97 1−2.48T+97T2 1 - 2.48T + 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

−9.060706973614027918032371588822, −8.085401829052476169004605324907, −6.94920502701385306542797534032, −6.17695662403297948521129459809, −5.83203985676142917041453773311, −4.89925993194376337297409156597, −4.16137327840218980552045591828, −3.42665088049623629573086463626, −2.07835637568624971126478106384, −0.908061193751690879881029934969, 0.908061193751690879881029934969, 2.07835637568624971126478106384, 3.42665088049623629573086463626, 4.16137327840218980552045591828, 4.89925993194376337297409156597, 5.83203985676142917041453773311, 6.17695662403297948521129459809, 6.94920502701385306542797534032, 8.085401829052476169004605324907, 9.060706973614027918032371588822

Graph of the ZZ-function along the critical line