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Properties

Label 2-1083-1.1-c1-0-3
Degree 22
Conductor 10831083
Sign 11
Analytic cond. 8.647798.64779
Root an. cond. 2.940712.94071
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  − 0.571·2-s + 3-s − 1.67·4-s − 2.67·5-s − 0.571·6-s − 3.67·7-s + 2.10·8-s + 9-s + 1.52·10-s − 3.81·11-s − 1.67·12-s + 0.143·13-s + 2.10·14-s − 2.67·15-s + 2.14·16-s − 0.571·18-s + 4.47·20-s − 3.67·21-s + 2.18·22-s + 7.52·23-s + 2.10·24-s + 2.14·25-s − 0.0823·26-s + 27-s + 6.14·28-s − 5.34·29-s + 1.52·30-s + ⋯
L(s)  = 1  − 0.404·2-s + 0.577·3-s − 0.836·4-s − 1.19·5-s − 0.233·6-s − 1.38·7-s + 0.742·8-s + 0.333·9-s + 0.483·10-s − 1.15·11-s − 0.482·12-s + 0.0399·13-s + 0.561·14-s − 0.690·15-s + 0.535·16-s − 0.134·18-s + 0.999·20-s − 0.801·21-s + 0.465·22-s + 1.56·23-s + 0.428·24-s + 0.428·25-s − 0.0161·26-s + 0.192·27-s + 1.16·28-s − 0.992·29-s + 0.279·30-s + ⋯

Functional equation

Λ(s)=(1083s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1083s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10831083    =    3⋅1923 \cdot 19^{2}
Sign: 11
Analytic conductor: 8.647798.64779
Root analytic conductor: 2.940712.94071
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1083, ( :1/2), 1)(2,\ 1083,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.61924556770.6192455677
L(12)L(\frac12) ≈\approx 0.61924556770.6192455677
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
19 1 1
good2 1+0.571T+2T2 1 + 0.571T + 2T^{2}
5 1+2.67T+5T2 1 + 2.67T + 5T^{2}
7 1+3.67T+7T2 1 + 3.67T + 7T^{2}
11 1+3.81T+11T2 1 + 3.81T + 11T^{2}
13 1−0.143T+13T2 1 - 0.143T + 13T^{2}
17 1+17T2 1 + 17T^{2}
23 1−7.52T+23T2 1 - 7.52T + 23T^{2}
29 1+5.34T+29T2 1 + 5.34T + 29T^{2}
31 1−8.81T+31T2 1 - 8.81T + 31T^{2}
37 1+T+37T2 1 + T + 37T^{2}
41 1−5.34T+41T2 1 - 5.34T + 41T^{2}
43 1−2.81T+43T2 1 - 2.81T + 43T^{2}
47 1+6T+47T2 1 + 6T + 47T^{2}
53 1−8.01T+53T2 1 - 8.01T + 53T^{2}
59 1−3.81T+59T2 1 - 3.81T + 59T^{2}
61 1−11.4T+61T2 1 - 11.4T + 61T^{2}
67 1−5.38T+67T2 1 - 5.38T + 67T^{2}
71 1+13.6T+71T2 1 + 13.6T + 71T^{2}
73 1+0.345T+73T2 1 + 0.345T + 73T^{2}
79 1−6.52T+79T2 1 - 6.52T + 79T^{2}
83 1+2.28T+83T2 1 + 2.28T + 83T^{2}
89 1+8.67T+89T2 1 + 8.67T + 89T^{2}
97 1+5.91T+97T2 1 + 5.91T + 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

−9.772177140935082487427771238511, −9.000219442929019549236696144475, −8.283619430828854722938608339671, −7.59236858195974866173836410043, −6.85775037422429070383656355746, −5.48178989383709369810953620445, −4.43637026507031162514883286637, −3.58097497061188803462347285515, −2.77759438759179916673741687981, −0.60456123115622303950618124783, 0.60456123115622303950618124783, 2.77759438759179916673741687981, 3.58097497061188803462347285515, 4.43637026507031162514883286637, 5.48178989383709369810953620445, 6.85775037422429070383656355746, 7.59236858195974866173836410043, 8.283619430828854722938608339671, 9.000219442929019549236696144475, 9.772177140935082487427771238511

Graph of the ZZ-function along the critical line