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Properties

Label 2-294-1.1-c5-0-17
Degree 22
Conductor 294294
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 9·3-s + 16·4-s − 4.50·5-s + 36·6-s − 64·8-s + 81·9-s + 18.0·10-s − 116.·11-s − 144·12-s + 85.4·13-s + 40.5·15-s + 256·16-s − 33.2·17-s − 324·18-s − 635.·19-s − 72.0·20-s + 464.·22-s + 2.72e3·23-s + 576·24-s − 3.10e3·25-s − 341.·26-s − 729·27-s + 5.86e3·29-s − 162.·30-s − 279.·31-s − 1.02e3·32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.0805·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.0569·10-s − 0.289·11-s − 0.288·12-s + 0.140·13-s + 0.0465·15-s + 0.250·16-s − 0.0279·17-s − 0.235·18-s − 0.403·19-s − 0.0402·20-s + 0.204·22-s + 1.07·23-s + 0.204·24-s − 0.993·25-s − 0.0991·26-s − 0.192·27-s + 1.29·29-s − 0.0328·30-s − 0.0521·31-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: −1-1
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: 11
Selberg data: (2, 294, ( :5/2), −1)(2,\ 294,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
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+4.50T+3.12e3T2 1 + 4.50T + 3.12e3T^{2}
11 1+116.T+1.61e5T2 1 + 116.T + 1.61e5T^{2}
13 1−85.4T+3.71e5T2 1 - 85.4T + 3.71e5T^{2}
17 1+33.2T+1.41e6T2 1 + 33.2T + 1.41e6T^{2}
19 1+635.T+2.47e6T2 1 + 635.T + 2.47e6T^{2}
23 1−2.72e3T+6.43e6T2 1 - 2.72e3T + 6.43e6T^{2}
29 1−5.86e3T+2.05e7T2 1 - 5.86e3T + 2.05e7T^{2}
31 1+279.T+2.86e7T2 1 + 279.T + 2.86e7T^{2}
37 1−3.03e3T+6.93e7T2 1 - 3.03e3T + 6.93e7T^{2}
41 1+819.T+1.15e8T2 1 + 819.T + 1.15e8T^{2}
43 1−1.11e4T+1.47e8T2 1 - 1.11e4T + 1.47e8T^{2}
47 1−7.40e3T+2.29e8T2 1 - 7.40e3T + 2.29e8T^{2}
53 1+1.36e4T+4.18e8T2 1 + 1.36e4T + 4.18e8T^{2}
59 1+2.23e4T+7.14e8T2 1 + 2.23e4T + 7.14e8T^{2}
61 1−1.26e4T+8.44e8T2 1 - 1.26e4T + 8.44e8T^{2}
67 1+5.23e4T+1.35e9T2 1 + 5.23e4T + 1.35e9T^{2}
71 1−6.02e4T+1.80e9T2 1 - 6.02e4T + 1.80e9T^{2}
73 1+7.69e4T+2.07e9T2 1 + 7.69e4T + 2.07e9T^{2}
79 1+3.35e4T+3.07e9T2 1 + 3.35e4T + 3.07e9T^{2}
83 1+6.05e4T+3.93e9T2 1 + 6.05e4T + 3.93e9T^{2}
89 1+9.21e4T+5.58e9T2 1 + 9.21e4T + 5.58e9T^{2}
97 1+1.52e5T+8.58e9T2 1 + 1.52e5T + 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

−10.51162765321645364134504111242, −9.586180665313000957081469432667, −8.573375545485203626352289805638, −7.58062582538427186700087087464, −6.60382439829072339785833393169, −5.60757472547168797879158001429, −4.33154866655228985853842867330, −2.76493769847232943658954962019, −1.27541090696420692534347657320, 0, 1.27541090696420692534347657320, 2.76493769847232943658954962019, 4.33154866655228985853842867330, 5.60757472547168797879158001429, 6.60382439829072339785833393169, 7.58062582538427186700087087464, 8.573375545485203626352289805638, 9.586180665313000957081469432667, 10.51162765321645364134504111242

Graph of the ZZ-function along the critical line