↑

Properties

Label 2-5225-1.1-c1-0-206
Degree 22
Conductor 52255225
Sign 11
Analytic cond. 41.721841.7218
Root an. cond. 6.459246.45924
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  + 2.40·2-s + 1.94·3-s + 3.78·4-s + 4.68·6-s − 0.665·7-s + 4.28·8-s + 0.794·9-s + 11-s + 7.36·12-s + 3.05·13-s − 1.59·14-s + 2.73·16-s + 4.49·17-s + 1.91·18-s + 19-s − 1.29·21-s + 2.40·22-s + 0.205·23-s + 8.34·24-s + 7.34·26-s − 4.29·27-s − 2.51·28-s + 0.0417·29-s + 6.76·31-s − 1.98·32-s + 1.94·33-s + 10.8·34-s + ⋯
L(s)  = 1  + 1.70·2-s + 1.12·3-s + 1.89·4-s + 1.91·6-s − 0.251·7-s + 1.51·8-s + 0.264·9-s + 0.301·11-s + 2.12·12-s + 0.847·13-s − 0.427·14-s + 0.683·16-s + 1.08·17-s + 0.450·18-s + 0.229·19-s − 0.282·21-s + 0.512·22-s + 0.0427·23-s + 1.70·24-s + 1.44·26-s − 0.826·27-s − 0.475·28-s + 0.00776·29-s + 1.21·31-s − 0.351·32-s + 0.339·33-s + 1.85·34-s + ⋯

Functional equation

Λ(s)=(5225s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5225s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 52255225    =    52⋅11⋅195^{2} \cdot 11 \cdot 19
Sign: 11
Analytic conductor: 41.721841.7218
Root analytic conductor: 6.459246.45924
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5225, ( :1/2), 1)(2,\ 5225,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 8.7116030138.711603013
L(12)L(\frac12) ≈\approx 8.7116030138.711603013
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)
bad5 1 1
11 1−T 1 - T
19 1−T 1 - T
good2 1−2.40T+2T2 1 - 2.40T + 2T^{2}
3 1−1.94T+3T2 1 - 1.94T + 3T^{2}
7 1+0.665T+7T2 1 + 0.665T + 7T^{2}
13 1−3.05T+13T2 1 - 3.05T + 13T^{2}
17 1−4.49T+17T2 1 - 4.49T + 17T^{2}
23 1−0.205T+23T2 1 - 0.205T + 23T^{2}
29 1−0.0417T+29T2 1 - 0.0417T + 29T^{2}
31 1−6.76T+31T2 1 - 6.76T + 31T^{2}
37 1−7.60T+37T2 1 - 7.60T + 37T^{2}
41 1−5.90T+41T2 1 - 5.90T + 41T^{2}
43 1+1.77T+43T2 1 + 1.77T + 43T^{2}
47 1+2.44T+47T2 1 + 2.44T + 47T^{2}
53 1+3.34T+53T2 1 + 3.34T + 53T^{2}
59 1−7.44T+59T2 1 - 7.44T + 59T^{2}
61 1+11.7T+61T2 1 + 11.7T + 61T^{2}
67 1+7.55T+67T2 1 + 7.55T + 67T^{2}
71 1−2.36T+71T2 1 - 2.36T + 71T^{2}
73 1+1.57T+73T2 1 + 1.57T + 73T^{2}
79 1+4.66T+79T2 1 + 4.66T + 79T^{2}
83 1−12.5T+83T2 1 - 12.5T + 83T^{2}
89 1−6.06T+89T2 1 - 6.06T + 89T^{2}
97 1−8.44T+97T2 1 - 8.44T + 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

−7.960490005246633530200480165702, −7.51758806371555721984659380455, −6.37872626274492668439207894176, −6.09401848432568600196876795821, −5.16987283543241482642315138717, −4.34288478887783214877913549269, −3.61080796841016430494597906193, −3.11313160669686981201942092390, −2.44870774196983349879227527548, −1.31796405591483271314376179127, 1.31796405591483271314376179127, 2.44870774196983349879227527548, 3.11313160669686981201942092390, 3.61080796841016430494597906193, 4.34288478887783214877913549269, 5.16987283543241482642315138717, 6.09401848432568600196876795821, 6.37872626274492668439207894176, 7.51758806371555721984659380455, 7.960490005246633530200480165702

Graph of the ZZ-function along the critical line