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Properties

Label 2-294-1.1-c5-0-18
Degree 22
Conductor 294294
Sign 11
Analytic cond. 47.152847.1528
Root an. cond. 6.866796.86679
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 9·3-s + 16·4-s + 75.4·5-s + 36·6-s + 64·8-s + 81·9-s + 301.·10-s − 149.·11-s + 144·12-s − 349.·13-s + 679.·15-s + 256·16-s + 1.14e3·17-s + 324·18-s + 2.79e3·19-s + 1.20e3·20-s − 597.·22-s + 1.81e3·23-s + 576·24-s + 2.57e3·25-s − 1.39e3·26-s + 729·27-s − 759.·29-s + 2.71e3·30-s − 9.03e3·31-s + 1.02e3·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.35·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.954·10-s − 0.372·11-s + 0.288·12-s − 0.573·13-s + 0.779·15-s + 0.250·16-s + 0.964·17-s + 0.235·18-s + 1.77·19-s + 0.675·20-s − 0.263·22-s + 0.715·23-s + 0.204·24-s + 0.823·25-s − 0.405·26-s + 0.192·27-s − 0.167·29-s + 0.551·30-s − 1.68·31-s + 0.176·32-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: 11
Analytic conductor: 47.152847.1528
Root analytic conductor: 6.866796.86679
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 294, ( :5/2), 1)(2,\ 294,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 5.4440272725.444027272
L(12)L(\frac12) ≈\approx 5.4440272725.444027272
L(72)L(\frac{7}{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 1−4T 1 - 4T
3 1−9T 1 - 9T
7 1 1
good5 1−75.4T+3.12e3T2 1 - 75.4T + 3.12e3T^{2}
11 1+149.T+1.61e5T2 1 + 149.T + 1.61e5T^{2}
13 1+349.T+3.71e5T2 1 + 349.T + 3.71e5T^{2}
17 1−1.14e3T+1.41e6T2 1 - 1.14e3T + 1.41e6T^{2}
19 1−2.79e3T+2.47e6T2 1 - 2.79e3T + 2.47e6T^{2}
23 1−1.81e3T+6.43e6T2 1 - 1.81e3T + 6.43e6T^{2}
29 1+759.T+2.05e7T2 1 + 759.T + 2.05e7T^{2}
31 1+9.03e3T+2.86e7T2 1 + 9.03e3T + 2.86e7T^{2}
37 1−7.79e3T+6.93e7T2 1 - 7.79e3T + 6.93e7T^{2}
41 1+7.64e3T+1.15e8T2 1 + 7.64e3T + 1.15e8T^{2}
43 1−1.21e4T+1.47e8T2 1 - 1.21e4T + 1.47e8T^{2}
47 1+2.45e4T+2.29e8T2 1 + 2.45e4T + 2.29e8T^{2}
53 1−1.35e4T+4.18e8T2 1 - 1.35e4T + 4.18e8T^{2}
59 1−2.63e4T+7.14e8T2 1 - 2.63e4T + 7.14e8T^{2}
61 1+3.53e4T+8.44e8T2 1 + 3.53e4T + 8.44e8T^{2}
67 1−5.43e4T+1.35e9T2 1 - 5.43e4T + 1.35e9T^{2}
71 1+7.01e4T+1.80e9T2 1 + 7.01e4T + 1.80e9T^{2}
73 1−4.44e4T+2.07e9T2 1 - 4.44e4T + 2.07e9T^{2}
79 1−6.16e4T+3.07e9T2 1 - 6.16e4T + 3.07e9T^{2}
83 1−8.71e4T+3.93e9T2 1 - 8.71e4T + 3.93e9T^{2}
89 1+9.85e4T+5.58e9T2 1 + 9.85e4T + 5.58e9T^{2}
97 1+3.23e4T+8.58e9T2 1 + 3.23e4T + 8.58e9T^{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

−10.90188447971699712849803674165, −9.807158247010878719365031544446, −9.373660577129236932390163105811, −7.86212993282366863671818867780, −6.98735982932894800674244015423, −5.66837638939159672045978117534, −5.09481266461612846990856491548, −3.44863478789883915223006110258, −2.48038006936622756390708073092, −1.32330637735012584549227107731, 1.32330637735012584549227107731, 2.48038006936622756390708073092, 3.44863478789883915223006110258, 5.09481266461612846990856491548, 5.66837638939159672045978117534, 6.98735982932894800674244015423, 7.86212993282366863671818867780, 9.373660577129236932390163105811, 9.807158247010878719365031544446, 10.90188447971699712849803674165

Graph of the ZZ-function along the critical line