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Properties

Label 2-294-1.1-c5-0-30
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 − 61.0·5-s + 36·6-s + 64·8-s + 81·9-s − 244.·10-s − 36.5·11-s + 144·12-s + 34.5·13-s − 549.·15-s + 256·16-s − 2.06e3·17-s + 324·18-s + 452.·19-s − 977.·20-s − 146.·22-s + 1.68e3·23-s + 576·24-s + 604.·25-s + 138.·26-s + 729·27-s − 4.76e3·29-s − 2.19e3·30-s − 5.26e3·31-s + 1.02e3·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.09·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.772·10-s − 0.0910·11-s + 0.288·12-s + 0.0567·13-s − 0.630·15-s + 0.250·16-s − 1.72·17-s + 0.235·18-s + 0.287·19-s − 0.546·20-s − 0.0643·22-s + 0.663·23-s + 0.204·24-s + 0.193·25-s + 0.0401·26-s + 0.192·27-s − 1.05·29-s − 0.446·30-s − 0.983·31-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+61.0T+3.12e3T2 1 + 61.0T + 3.12e3T^{2}
11 1+36.5T+1.61e5T2 1 + 36.5T + 1.61e5T^{2}
13 1−34.5T+3.71e5T2 1 - 34.5T + 3.71e5T^{2}
17 1+2.06e3T+1.41e6T2 1 + 2.06e3T + 1.41e6T^{2}
19 1−452.T+2.47e6T2 1 - 452.T + 2.47e6T^{2}
23 1−1.68e3T+6.43e6T2 1 - 1.68e3T + 6.43e6T^{2}
29 1+4.76e3T+2.05e7T2 1 + 4.76e3T + 2.05e7T^{2}
31 1+5.26e3T+2.86e7T2 1 + 5.26e3T + 2.86e7T^{2}
37 1+1.28e4T+6.93e7T2 1 + 1.28e4T + 6.93e7T^{2}
41 1+7.12e3T+1.15e8T2 1 + 7.12e3T + 1.15e8T^{2}
43 1−1.11e4T+1.47e8T2 1 - 1.11e4T + 1.47e8T^{2}
47 1+2.34e4T+2.29e8T2 1 + 2.34e4T + 2.29e8T^{2}
53 1+7.03e3T+4.18e8T2 1 + 7.03e3T + 4.18e8T^{2}
59 1+4.42e4T+7.14e8T2 1 + 4.42e4T + 7.14e8T^{2}
61 1−1.93e4T+8.44e8T2 1 - 1.93e4T + 8.44e8T^{2}
67 1−2.09e4T+1.35e9T2 1 - 2.09e4T + 1.35e9T^{2}
71 1−7.98e4T+1.80e9T2 1 - 7.98e4T + 1.80e9T^{2}
73 1+3.70e4T+2.07e9T2 1 + 3.70e4T + 2.07e9T^{2}
79 1−4.20e4T+3.07e9T2 1 - 4.20e4T + 3.07e9T^{2}
83 1+6.31e3T+3.93e9T2 1 + 6.31e3T + 3.93e9T^{2}
89 1+5.14e4T+5.58e9T2 1 + 5.14e4T + 5.58e9T^{2}
97 1+1.27e5T+8.58e9T2 1 + 1.27e5T + 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

−10.84020485050575768882622092704, −9.379939722301625998558016667219, −8.433418661647511944915712787030, −7.45647517156981617631874505979, −6.66529723581969905345452290922, −5.14032567574772662200908184455, −4.10284479977844316988881523722, −3.26979572967773874518544459932, −1.90902669746424570415111784594, 0, 1.90902669746424570415111784594, 3.26979572967773874518544459932, 4.10284479977844316988881523722, 5.14032567574772662200908184455, 6.66529723581969905345452290922, 7.45647517156981617631874505979, 8.433418661647511944915712787030, 9.379939722301625998558016667219, 10.84020485050575768882622092704

Graph of the ZZ-function along the critical line