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Properties

Label 2-1881-1.1-c1-0-1
Degree 22
Conductor 18811881
Sign 11
Analytic cond. 15.019815.0198
Root an. cond. 3.875543.87554
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  − 1.45·2-s + 0.114·4-s − 2.59·5-s − 2.00·7-s + 2.74·8-s + 3.77·10-s + 11-s − 4.45·13-s + 2.92·14-s − 4.21·16-s − 4.54·17-s + 19-s − 0.297·20-s − 1.45·22-s − 7.48·23-s + 1.72·25-s + 6.48·26-s − 0.230·28-s + 3.17·29-s + 9.34·31-s + 0.647·32-s + 6.60·34-s + 5.21·35-s − 6.84·37-s − 1.45·38-s − 7.10·40-s − 0.644·41-s + ⋯
L(s)  = 1  − 1.02·2-s + 0.0572·4-s − 1.15·5-s − 0.759·7-s + 0.969·8-s + 1.19·10-s + 0.301·11-s − 1.23·13-s + 0.780·14-s − 1.05·16-s − 1.10·17-s + 0.229·19-s − 0.0664·20-s − 0.310·22-s − 1.56·23-s + 0.344·25-s + 1.27·26-s − 0.0435·28-s + 0.589·29-s + 1.67·31-s + 0.114·32-s + 1.13·34-s + 0.880·35-s − 1.12·37-s − 0.235·38-s − 1.12·40-s − 0.100·41-s + ⋯

Functional equation

Λ(s)=(1881s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1881s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18811881    =    32⋅11⋅193^{2} \cdot 11 \cdot 19
Sign: 11
Analytic conductor: 15.019815.0198
Root analytic conductor: 3.875543.87554
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1881, ( :1/2), 1)(2,\ 1881,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.24769885960.2476988596
L(12)L(\frac12) ≈\approx 0.24769885960.2476988596
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 1
11 1−T 1 - T
19 1−T 1 - T
good2 1+1.45T+2T2 1 + 1.45T + 2T^{2}
5 1+2.59T+5T2 1 + 2.59T + 5T^{2}
7 1+2.00T+7T2 1 + 2.00T + 7T^{2}
13 1+4.45T+13T2 1 + 4.45T + 13T^{2}
17 1+4.54T+17T2 1 + 4.54T + 17T^{2}
23 1+7.48T+23T2 1 + 7.48T + 23T^{2}
29 1−3.17T+29T2 1 - 3.17T + 29T^{2}
31 1−9.34T+31T2 1 - 9.34T + 31T^{2}
37 1+6.84T+37T2 1 + 6.84T + 37T^{2}
41 1+0.644T+41T2 1 + 0.644T + 41T^{2}
43 1+8.07T+43T2 1 + 8.07T + 43T^{2}
47 1+11.6T+47T2 1 + 11.6T + 47T^{2}
53 1−5.53T+53T2 1 - 5.53T + 53T^{2}
59 1+9.16T+59T2 1 + 9.16T + 59T^{2}
61 1−9.45T+61T2 1 - 9.45T + 61T^{2}
67 1−0.113T+67T2 1 - 0.113T + 67T^{2}
71 1−9.84T+71T2 1 - 9.84T + 71T^{2}
73 1+2.38T+73T2 1 + 2.38T + 73T^{2}
79 1+2.01T+79T2 1 + 2.01T + 79T^{2}
83 1−2.90T+83T2 1 - 2.90T + 83T^{2}
89 1−8.82T+89T2 1 - 8.82T + 89T^{2}
97 1−11.0T+97T2 1 - 11.0T + 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

−9.236354673463371500147588553314, −8.307674940757123310161751973296, −7.973147994341432246150453209501, −7.00387542715732833414903504755, −6.46741124524642745373776241764, −4.93098622258183975261681190345, −4.29953245786613256529485147732, −3.32987825402886191946760069002, −2.03200904359066458909134711951, −0.37616945552612121761066263649, 0.37616945552612121761066263649, 2.03200904359066458909134711951, 3.32987825402886191946760069002, 4.29953245786613256529485147732, 4.93098622258183975261681190345, 6.46741124524642745373776241764, 7.00387542715732833414903504755, 7.973147994341432246150453209501, 8.307674940757123310161751973296, 9.236354673463371500147588553314

Graph of the ZZ-function along the critical line