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Properties

Label 2-7942-1.1-c1-0-50
Degree 22
Conductor 79427942
Sign 11
Analytic cond. 63.417163.4171
Root an. cond. 7.963497.96349
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-s + 0.337·3-s + 4-s − 2.62·5-s + 0.337·6-s − 0.195·7-s + 8-s − 2.88·9-s − 2.62·10-s − 11-s + 0.337·12-s + 2.97·13-s − 0.195·14-s − 0.886·15-s + 16-s + 0.422·17-s − 2.88·18-s − 2.62·20-s − 0.0659·21-s − 22-s − 1.65·23-s + 0.337·24-s + 1.90·25-s + 2.97·26-s − 1.98·27-s − 0.195·28-s − 2.77·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.194·3-s + 0.5·4-s − 1.17·5-s + 0.137·6-s − 0.0739·7-s + 0.353·8-s − 0.962·9-s − 0.831·10-s − 0.301·11-s + 0.0974·12-s + 0.823·13-s − 0.0522·14-s − 0.228·15-s + 0.250·16-s + 0.102·17-s − 0.680·18-s − 0.587·20-s − 0.0143·21-s − 0.213·22-s − 0.344·23-s + 0.0688·24-s + 0.381·25-s + 0.582·26-s − 0.382·27-s − 0.0369·28-s − 0.515·29-s + ⋯

Functional equation

Λ(s)=(7942s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7942s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 79427942    =    2⋅11⋅1922 \cdot 11 \cdot 19^{2}
Sign: 11
Analytic conductor: 63.417163.4171
Root analytic conductor: 7.963497.96349
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7942, ( :1/2), 1)(2,\ 7942,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.0728676352.072867635
L(12)L(\frac12) ≈\approx 2.0728676352.072867635
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)
bad2 1−T 1 - T
11 1+T 1 + T
19 1 1
good3 1−0.337T+3T2 1 - 0.337T + 3T^{2}
5 1+2.62T+5T2 1 + 2.62T + 5T^{2}
7 1+0.195T+7T2 1 + 0.195T + 7T^{2}
13 1−2.97T+13T2 1 - 2.97T + 13T^{2}
17 1−0.422T+17T2 1 - 0.422T + 17T^{2}
23 1+1.65T+23T2 1 + 1.65T + 23T^{2}
29 1+2.77T+29T2 1 + 2.77T + 29T^{2}
31 1+7.95T+31T2 1 + 7.95T + 31T^{2}
37 1−9.69T+37T2 1 - 9.69T + 37T^{2}
41 1−6.33T+41T2 1 - 6.33T + 41T^{2}
43 1+0.285T+43T2 1 + 0.285T + 43T^{2}
47 1−1.99T+47T2 1 - 1.99T + 47T^{2}
53 1−3.41T+53T2 1 - 3.41T + 53T^{2}
59 1−14.2T+59T2 1 - 14.2T + 59T^{2}
61 1+8.39T+61T2 1 + 8.39T + 61T^{2}
67 1−6.72T+67T2 1 - 6.72T + 67T^{2}
71 1−1.72T+71T2 1 - 1.72T + 71T^{2}
73 1+13.5T+73T2 1 + 13.5T + 73T^{2}
79 1−7.36T+79T2 1 - 7.36T + 79T^{2}
83 1−7.03T+83T2 1 - 7.03T + 83T^{2}
89 1−9.88T+89T2 1 - 9.88T + 89T^{2}
97 1−10.0T+97T2 1 - 10.0T + 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

−7.75615832692780123141436001480, −7.30170968072111324991315176285, −6.26079994231531393397580269645, −5.78583216582486899183176645164, −5.00498540259305226646251136468, −4.06848057017784374830794137222, −3.65193470312330684018686686090, −2.92195980755777667540263923359, −2.02034026097643827379544910314, −0.61172772485709602035310399327, 0.61172772485709602035310399327, 2.02034026097643827379544910314, 2.92195980755777667540263923359, 3.65193470312330684018686686090, 4.06848057017784374830794137222, 5.00498540259305226646251136468, 5.78583216582486899183176645164, 6.26079994231531393397580269645, 7.30170968072111324991315176285, 7.75615832692780123141436001480

Graph of the ZZ-function along the critical line