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Properties

Label 2-7942-1.1-c1-0-141
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 − 3.11·3-s + 4-s + 2.93·5-s − 3.11·6-s + 4.25·7-s + 8-s + 6.67·9-s + 2.93·10-s − 11-s − 3.11·12-s + 4.02·13-s + 4.25·14-s − 9.11·15-s + 16-s − 3.10·17-s + 6.67·18-s + 2.93·20-s − 13.2·21-s − 22-s + 8.63·23-s − 3.11·24-s + 3.59·25-s + 4.02·26-s − 11.4·27-s + 4.25·28-s + 1.90·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.79·3-s + 0.5·4-s + 1.31·5-s − 1.27·6-s + 1.60·7-s + 0.353·8-s + 2.22·9-s + 0.926·10-s − 0.301·11-s − 0.898·12-s + 1.11·13-s + 1.13·14-s − 2.35·15-s + 0.250·16-s − 0.753·17-s + 1.57·18-s + 0.655·20-s − 2.89·21-s − 0.213·22-s + 1.79·23-s − 0.635·24-s + 0.718·25-s + 0.789·26-s − 2.20·27-s + 0.804·28-s + 0.354·29-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 3.3450894563.345089456
L(12)L(\frac12) ≈\approx 3.3450894563.345089456
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+3.11T+3T2 1 + 3.11T + 3T^{2}
5 1−2.93T+5T2 1 - 2.93T + 5T^{2}
7 1−4.25T+7T2 1 - 4.25T + 7T^{2}
13 1−4.02T+13T2 1 - 4.02T + 13T^{2}
17 1+3.10T+17T2 1 + 3.10T + 17T^{2}
23 1−8.63T+23T2 1 - 8.63T + 23T^{2}
29 1−1.90T+29T2 1 - 1.90T + 29T^{2}
31 1+1.66T+31T2 1 + 1.66T + 31T^{2}
37 1+6.62T+37T2 1 + 6.62T + 37T^{2}
41 1−4.69T+41T2 1 - 4.69T + 41T^{2}
43 1−3.02T+43T2 1 - 3.02T + 43T^{2}
47 1−5.87T+47T2 1 - 5.87T + 47T^{2}
53 1−0.147T+53T2 1 - 0.147T + 53T^{2}
59 1−4.30T+59T2 1 - 4.30T + 59T^{2}
61 1+7.53T+61T2 1 + 7.53T + 61T^{2}
67 1−2.17T+67T2 1 - 2.17T + 67T^{2}
71 1−13.7T+71T2 1 - 13.7T + 71T^{2}
73 1+10.8T+73T2 1 + 10.8T + 73T^{2}
79 1+0.921T+79T2 1 + 0.921T + 79T^{2}
83 1−0.684T+83T2 1 - 0.684T + 83T^{2}
89 1+13.4T+89T2 1 + 13.4T + 89T^{2}
97 1−16.7T+97T2 1 - 16.7T + 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

−7.45404867859389590065929304129, −6.88592967640019859540239210658, −6.12924539477425536558930468915, −5.72388173496940269041875899274, −5.01692057585122419999060079488, −4.80782657666912414391825377325, −3.87159687893510848468167550197, −2.45516351770649981697556762127, −1.58150768126396126877069543429, −1.01273170111468682354955230828, 1.01273170111468682354955230828, 1.58150768126396126877069543429, 2.45516351770649981697556762127, 3.87159687893510848468167550197, 4.80782657666912414391825377325, 5.01692057585122419999060079488, 5.72388173496940269041875899274, 6.12924539477425536558930468915, 6.88592967640019859540239210658, 7.45404867859389590065929304129

Graph of the ZZ-function along the critical line