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Properties

Label 2-294-1.1-c7-0-19
Degree 22
Conductor 294294
Sign 11
Analytic cond. 91.841191.8411
Root an. cond. 9.583389.58338
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 27·3-s + 64·4-s + 18·5-s − 216·6-s − 512·8-s + 729·9-s − 144·10-s + 8.17e3·11-s + 1.72e3·12-s + 1.42e4·13-s + 486·15-s + 4.09e3·16-s + 2.14e4·17-s − 5.83e3·18-s + 5.88e3·19-s + 1.15e3·20-s − 6.53e4·22-s − 9.87e4·23-s − 1.38e4·24-s − 7.78e4·25-s − 1.13e5·26-s + 1.96e4·27-s + 1.65e5·29-s − 3.88e3·30-s + 2.41e5·31-s − 3.27e4·32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.0643·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.0455·10-s + 1.85·11-s + 0.288·12-s + 1.79·13-s + 0.0371·15-s + 1/4·16-s + 1.05·17-s − 0.235·18-s + 0.196·19-s + 0.0321·20-s − 1.30·22-s − 1.69·23-s − 0.204·24-s − 0.995·25-s − 1.27·26-s + 0.192·27-s + 1.25·29-s − 0.0262·30-s + 1.45·31-s − 0.176·32-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+7/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+7/2) \, 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: 91.841191.8411
Root analytic conductor: 9.583389.58338
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 294, ( :7/2), 1)(2,\ 294,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 2.7206682602.720668260
L(12)L(\frac12) ≈\approx 2.7206682602.720668260
L(92)L(\frac{9}{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+p3T 1 + p^{3} T
3 1−p3T 1 - p^{3} T
7 1 1
good5 1−18T+p7T2 1 - 18 T + p^{7} T^{2}
11 1−8172T+p7T2 1 - 8172 T + p^{7} T^{2}
13 1−14242T+p7T2 1 - 14242 T + p^{7} T^{2}
17 1−21462T+p7T2 1 - 21462 T + p^{7} T^{2}
19 1−5884T+p7T2 1 - 5884 T + p^{7} T^{2}
23 1+98784T+p7T2 1 + 98784 T + p^{7} T^{2}
29 1−165174T+p7T2 1 - 165174 T + p^{7} T^{2}
31 1−241312T+p7T2 1 - 241312 T + p^{7} T^{2}
37 1−185438T+p7T2 1 - 185438 T + p^{7} T^{2}
41 1+59682T+p7T2 1 + 59682 T + p^{7} T^{2}
43 1+809308T+p7T2 1 + 809308 T + p^{7} T^{2}
47 1+942096T+p7T2 1 + 942096 T + p^{7} T^{2}
53 1−226398T+p7T2 1 - 226398 T + p^{7} T^{2}
59 1−2205732T+p7T2 1 - 2205732 T + p^{7} T^{2}
61 1−1156690T+p7T2 1 - 1156690 T + p^{7} T^{2}
67 1+3740404T+p7T2 1 + 3740404 T + p^{7} T^{2}
71 1+2593296T+p7T2 1 + 2593296 T + p^{7} T^{2}
73 1−1038742T+p7T2 1 - 1038742 T + p^{7} T^{2}
79 1−2280032T+p7T2 1 - 2280032 T + p^{7} T^{2}
83 1−283404T+p7T2 1 - 283404 T + p^{7} T^{2}
89 1−5227230T+p7T2 1 - 5227230 T + p^{7} T^{2}
97 1+6168770T+p7T2 1 + 6168770 T + p^{7} T^{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

−10.20469563771074407083015823322, −9.625662773121024660849698049800, −8.524603203471837991108608791334, −8.078800172039724744198107843986, −6.63109656882668245115689214363, −6.02924582768812313915111455846, −4.11571939084227571707929134968, −3.30751852104742682794155727624, −1.70166152361807475035783182085, −0.976390682087626548943334349174, 0.976390682087626548943334349174, 1.70166152361807475035783182085, 3.30751852104742682794155727624, 4.11571939084227571707929134968, 6.02924582768812313915111455846, 6.63109656882668245115689214363, 8.078800172039724744198107843986, 8.524603203471837991108608791334, 9.625662773121024660849698049800, 10.20469563771074407083015823322

Graph of the ZZ-function along the critical line