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Properties

Label 2-55-1.1-c1-0-2
Degree 22
Conductor 5555
Sign 11
Analytic cond. 0.4391770.439177
Root an. cond. 0.6627040.662704
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.41·2-s − 2.82·3-s + 3.82·4-s − 5-s − 6.82·6-s − 2·7-s + 4.41·8-s + 5.00·9-s − 2.41·10-s + 11-s − 10.8·12-s − 1.17·13-s − 4.82·14-s + 2.82·15-s + 2.99·16-s + 6.82·17-s + 12.0·18-s − 3.82·20-s + 5.65·21-s + 2.41·22-s − 2.82·23-s − 12.4·24-s + 25-s − 2.82·26-s − 5.65·27-s − 7.65·28-s − 3.65·29-s + ⋯
L(s)  = 1  + 1.70·2-s − 1.63·3-s + 1.91·4-s − 0.447·5-s − 2.78·6-s − 0.755·7-s + 1.56·8-s + 1.66·9-s − 0.763·10-s + 0.301·11-s − 3.12·12-s − 0.324·13-s − 1.29·14-s + 0.730·15-s + 0.749·16-s + 1.65·17-s + 2.84·18-s − 0.856·20-s + 1.23·21-s + 0.514·22-s − 0.589·23-s − 2.54·24-s + 0.200·25-s − 0.554·26-s − 1.08·27-s − 1.44·28-s − 0.679·29-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: 11
Analytic conductor: 0.4391770.439177
Root analytic conductor: 0.6627040.662704
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 55, ( :1/2), 1)(2,\ 55,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1970569671.197056967
L(12)L(\frac12) ≈\approx 1.1970569671.197056967
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+T 1 + T
11 1−T 1 - T
good2 1−2.41T+2T2 1 - 2.41T + 2T^{2}
3 1+2.82T+3T2 1 + 2.82T + 3T^{2}
7 1+2T+7T2 1 + 2T + 7T^{2}
13 1+1.17T+13T2 1 + 1.17T + 13T^{2}
17 1−6.82T+17T2 1 - 6.82T + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1+2.82T+23T2 1 + 2.82T + 23T^{2}
29 1+3.65T+29T2 1 + 3.65T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+7.65T+37T2 1 + 7.65T + 37T^{2}
41 1−6T+41T2 1 - 6T + 41T^{2}
43 1+6T+43T2 1 + 6T + 43T^{2}
47 1−2.82T+47T2 1 - 2.82T + 47T^{2}
53 1−11.6T+53T2 1 - 11.6T + 53T^{2}
59 1−1.65T+59T2 1 - 1.65T + 59T^{2}
61 1+9.31T+61T2 1 + 9.31T + 61T^{2}
67 1−12.4T+67T2 1 - 12.4T + 67T^{2}
71 1−11.3T+71T2 1 - 11.3T + 71T^{2}
73 1+1.17T+73T2 1 + 1.17T + 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1+6T+83T2 1 + 6T + 83T^{2}
89 1+13.3T+89T2 1 + 13.3T + 89T^{2}
97 1−3.65T+97T2 1 - 3.65T + 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

−15.34960948708537437517013199535, −14.10678218029280958479780612667, −12.69871244487788176118942150891, −12.18106320759827562104741174082, −11.34576766095860460302436742769, −10.09200070663283635358165303574, −7.19557878469241887119343312891, −6.08854885975527628228900382813, −5.16733014703725630411671370919, −3.70347734990671907639955801345, 3.70347734990671907639955801345, 5.16733014703725630411671370919, 6.08854885975527628228900382813, 7.19557878469241887119343312891, 10.09200070663283635358165303574, 11.34576766095860460302436742769, 12.18106320759827562104741174082, 12.69871244487788176118942150891, 14.10678218029280958479780612667, 15.34960948708537437517013199535

Graph of the ZZ-function along the critical line