↑

Properties

Label 2-294-1.1-c5-0-31
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 9·3-s + 16·4-s − 46.9·5-s + 36·6-s + 64·8-s + 81·9-s − 187.·10-s − 87.4·11-s + 144·12-s − 754.·13-s − 422.·15-s + 256·16-s + 1.44e3·17-s + 324·18-s − 2.54e3·19-s − 750.·20-s − 349.·22-s − 912.·23-s + 576·24-s − 922.·25-s − 3.01e3·26-s + 729·27-s + 173.·29-s − 1.68e3·30-s − 4.53e3·31-s + 1.02e3·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.839·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.593·10-s − 0.217·11-s + 0.288·12-s − 1.23·13-s − 0.484·15-s + 0.250·16-s + 1.21·17-s + 0.235·18-s − 1.61·19-s − 0.419·20-s − 0.154·22-s − 0.359·23-s + 0.204·24-s − 0.295·25-s − 0.875·26-s + 0.192·27-s + 0.0382·29-s − 0.342·30-s − 0.846·31-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+46.9T+3.12e3T2 1 + 46.9T + 3.12e3T^{2}
11 1+87.4T+1.61e5T2 1 + 87.4T + 1.61e5T^{2}
13 1+754.T+3.71e5T2 1 + 754.T + 3.71e5T^{2}
17 1−1.44e3T+1.41e6T2 1 - 1.44e3T + 1.41e6T^{2}
19 1+2.54e3T+2.47e6T2 1 + 2.54e3T + 2.47e6T^{2}
23 1+912.T+6.43e6T2 1 + 912.T + 6.43e6T^{2}
29 1−173.T+2.05e7T2 1 - 173.T + 2.05e7T^{2}
31 1+4.53e3T+2.86e7T2 1 + 4.53e3T + 2.86e7T^{2}
37 1−6.82e3T+6.93e7T2 1 - 6.82e3T + 6.93e7T^{2}
41 1+1.30e4T+1.15e8T2 1 + 1.30e4T + 1.15e8T^{2}
43 1+1.22e4T+1.47e8T2 1 + 1.22e4T + 1.47e8T^{2}
47 1+1.34e4T+2.29e8T2 1 + 1.34e4T + 2.29e8T^{2}
53 1+9.67e3T+4.18e8T2 1 + 9.67e3T + 4.18e8T^{2}
59 1+3.03e4T+7.14e8T2 1 + 3.03e4T + 7.14e8T^{2}
61 1+732.T+8.44e8T2 1 + 732.T + 8.44e8T^{2}
67 1−4.63e4T+1.35e9T2 1 - 4.63e4T + 1.35e9T^{2}
71 1+3.29e3T+1.80e9T2 1 + 3.29e3T + 1.80e9T^{2}
73 1+1.02e4T+2.07e9T2 1 + 1.02e4T + 2.07e9T^{2}
79 1−9.85e4T+3.07e9T2 1 - 9.85e4T + 3.07e9T^{2}
83 1+8.77e4T+3.93e9T2 1 + 8.77e4T + 3.93e9T^{2}
89 1−6.90e4T+5.58e9T2 1 - 6.90e4T + 5.58e9T^{2}
97 1−4.21e4T+8.58e9T2 1 - 4.21e4T + 8.58e9T^{2}
show more
show less
   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

−10.51590376643020948724991736487, −9.618435821522000277883246917381, −8.222255341193597779137211559764, −7.63949462728791222179988627107, −6.56065676400068965456360293352, −5.17126059653199896645049647197, −4.15823314085556576211956936436, −3.18018357860610324492688567034, −1.95131346687695533031826424056, 0, 1.95131346687695533031826424056, 3.18018357860610324492688567034, 4.15823314085556576211956936436, 5.17126059653199896645049647197, 6.56065676400068965456360293352, 7.63949462728791222179988627107, 8.222255341193597779137211559764, 9.618435821522000277883246917381, 10.51590376643020948724991736487

Graph of the ZZ-function along the critical line