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Properties

Label 2-7942-1.1-c1-0-34
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 − 1.35·3-s + 4-s + 2.80·5-s − 1.35·6-s − 4.57·7-s + 8-s − 1.15·9-s + 2.80·10-s − 11-s − 1.35·12-s − 6.22·13-s − 4.57·14-s − 3.80·15-s + 16-s − 3.01·17-s − 1.15·18-s + 2.80·20-s + 6.20·21-s − 22-s + 7.65·23-s − 1.35·24-s + 2.84·25-s − 6.22·26-s + 5.64·27-s − 4.57·28-s + 5.20·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.783·3-s + 0.5·4-s + 1.25·5-s − 0.554·6-s − 1.72·7-s + 0.353·8-s − 0.385·9-s + 0.885·10-s − 0.301·11-s − 0.391·12-s − 1.72·13-s − 1.22·14-s − 0.981·15-s + 0.250·16-s − 0.731·17-s − 0.272·18-s + 0.626·20-s + 1.35·21-s − 0.213·22-s + 1.59·23-s − 0.277·24-s + 0.569·25-s − 1.22·26-s + 1.08·27-s − 0.864·28-s + 0.967·29-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 1.5014496471.501449647
L(12)L(\frac12) ≈\approx 1.5014496471.501449647
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+1.35T+3T2 1 + 1.35T + 3T^{2}
5 1−2.80T+5T2 1 - 2.80T + 5T^{2}
7 1+4.57T+7T2 1 + 4.57T + 7T^{2}
13 1+6.22T+13T2 1 + 6.22T + 13T^{2}
17 1+3.01T+17T2 1 + 3.01T + 17T^{2}
23 1−7.65T+23T2 1 - 7.65T + 23T^{2}
29 1−5.20T+29T2 1 - 5.20T + 29T^{2}
31 1+3.41T+31T2 1 + 3.41T + 31T^{2}
37 1+5.26T+37T2 1 + 5.26T + 37T^{2}
41 1+0.287T+41T2 1 + 0.287T + 41T^{2}
43 1+8.80T+43T2 1 + 8.80T + 43T^{2}
47 1−10.4T+47T2 1 - 10.4T + 47T^{2}
53 1−3.46T+53T2 1 - 3.46T + 53T^{2}
59 1+11.9T+59T2 1 + 11.9T + 59T^{2}
61 1+6.27T+61T2 1 + 6.27T + 61T^{2}
67 1+3.24T+67T2 1 + 3.24T + 67T^{2}
71 1−10.4T+71T2 1 - 10.4T + 71T^{2}
73 1−12.7T+73T2 1 - 12.7T + 73T^{2}
79 1−0.452T+79T2 1 - 0.452T + 79T^{2}
83 1−6.14T+83T2 1 - 6.14T + 83T^{2}
89 1+9.94T+89T2 1 + 9.94T + 89T^{2}
97 1−11.9T+97T2 1 - 11.9T + 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

−7.40443746781336380380819699924, −6.75681022404191875447377000563, −6.45098716721521792171709736010, −5.69424113233929225947488179074, −5.18323838747477789244855268949, −4.61999005723385196003913518259, −3.29336868240501289126136924529, −2.77326693599957691430189575212, −2.08253689663721159093604068489, −0.52694174490265709018471195572, 0.52694174490265709018471195572, 2.08253689663721159093604068489, 2.77326693599957691430189575212, 3.29336868240501289126136924529, 4.61999005723385196003913518259, 5.18323838747477789244855268949, 5.69424113233929225947488179074, 6.45098716721521792171709736010, 6.75681022404191875447377000563, 7.40443746781336380380819699924

Graph of the ZZ-function along the critical line