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Properties

Label 2-294-7.4-c5-0-25
Degree 22
Conductor 294294
Sign −0.386+0.922i-0.386 + 0.922i
Analytic cond. 47.152847.1528
Root an. cond. 6.866796.86679
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 3.46i)2-s + (4.5 − 7.79i)3-s + (−7.99 + 13.8i)4-s + (−3 − 5.19i)5-s − 36·6-s + 63.9·8-s + (−40.5 − 70.1i)9-s + (−12 + 20.7i)10-s + (333 − 576. i)11-s + (72 + 124. i)12-s + 559·13-s − 54·15-s + (−128 − 221. i)16-s + (−870 + 1.50e3i)17-s + (−162 + 280. i)18-s + (578.5 + 1.00e3i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.0536 − 0.0929i)5-s − 0.408·6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.0379 + 0.0657i)10-s + (0.829 − 1.43i)11-s + (0.144 + 0.249i)12-s + 0.917·13-s − 0.0619·15-s + (−0.125 − 0.216i)16-s + (−0.730 + 1.26i)17-s + (−0.117 + 0.204i)18-s + (0.367 + 0.636i)19-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=((−0.386+0.922i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+5/2) L(s)=((−0.386+0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: −0.386+0.922i-0.386 + 0.922i
Analytic conductor: 47.152847.1528
Root analytic conductor: 6.866796.86679
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ294(67,⋅)\chi_{294} (67, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 294, ( :5/2), −0.386+0.922i)(2,\ 294,\ (\ :5/2),\ -0.386 + 0.922i)

Particular Values

L(3)L(3) ≈\approx 1.9321795181.932179518
L(12)L(\frac12) ≈\approx 1.9321795181.932179518
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+(2+3.46i)T 1 + (2 + 3.46i)T
3 1+(−4.5+7.79i)T 1 + (-4.5 + 7.79i)T
7 1 1
good5 1+(3+5.19i)T+(−1.56e3+2.70e3i)T2 1 + (3 + 5.19i)T + (-1.56e3 + 2.70e3i)T^{2}
11 1+(−333+576.i)T+(−8.05e4−1.39e5i)T2 1 + (-333 + 576. i)T + (-8.05e4 - 1.39e5i)T^{2}
13 1−559T+3.71e5T2 1 - 559T + 3.71e5T^{2}
17 1+(870−1.50e3i)T+(−7.09e5−1.22e6i)T2 1 + (870 - 1.50e3i)T + (-7.09e5 - 1.22e6i)T^{2}
19 1+(−578.5−1.00e3i)T+(−1.23e6+2.14e6i)T2 1 + (-578.5 - 1.00e3i)T + (-1.23e6 + 2.14e6i)T^{2}
23 1+(−1.73e3−3.00e3i)T+(−3.21e6+5.57e6i)T2 1 + (-1.73e3 - 3.00e3i)T + (-3.21e6 + 5.57e6i)T^{2}
29 1−3.37e3T+2.05e7T2 1 - 3.37e3T + 2.05e7T^{2}
31 1+(−3.14e3+5.44e3i)T+(−1.43e7−2.47e7i)T2 1 + (-3.14e3 + 5.44e3i)T + (-1.43e7 - 2.47e7i)T^{2}
37 1+(1.56e3+2.71e3i)T+(−3.46e7+6.00e7i)T2 1 + (1.56e3 + 2.71e3i)T + (-3.46e7 + 6.00e7i)T^{2}
41 1−4.86e3T+1.15e8T2 1 - 4.86e3T + 1.15e8T^{2}
43 1+1.14e4T+1.47e8T2 1 + 1.14e4T + 1.47e8T^{2}
47 1+(−1.15e3−2.00e3i)T+(−1.14e8+1.98e8i)T2 1 + (-1.15e3 - 2.00e3i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(−1.41e4+2.45e4i)T+(−2.09e8−3.62e8i)T2 1 + (-1.41e4 + 2.45e4i)T + (-2.09e8 - 3.62e8i)T^{2}
59 1+(−1.02e4+1.77e4i)T+(−3.57e8−6.19e8i)T2 1 + (-1.02e4 + 1.77e4i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(2.31e3+4.00e3i)T+(−4.22e8+7.31e8i)T2 1 + (2.31e3 + 4.00e3i)T + (-4.22e8 + 7.31e8i)T^{2}
67 1+(−9.37e3+1.62e4i)T+(−6.75e8−1.16e9i)T2 1 + (-9.37e3 + 1.62e4i)T + (-6.75e8 - 1.16e9i)T^{2}
71 1+3.82e4T+1.80e9T2 1 + 3.82e4T + 1.80e9T^{2}
73 1+(−3.52e4+6.11e4i)T+(−1.03e9−1.79e9i)T2 1 + (-3.52e4 + 6.11e4i)T + (-1.03e9 - 1.79e9i)T^{2}
79 1+(−3.11e4−5.39e4i)T+(−1.53e9+2.66e9i)T2 1 + (-3.11e4 - 5.39e4i)T + (-1.53e9 + 2.66e9i)T^{2}
83 1+7.98e4T+3.93e9T2 1 + 7.98e4T + 3.93e9T^{2}
89 1+(9.06e3+1.56e4i)T+(−2.79e9+4.83e9i)T2 1 + (9.06e3 + 1.56e4i)T + (-2.79e9 + 4.83e9i)T^{2}
97 1+1.24e5T+8.58e9T2 1 + 1.24e5T + 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

−10.85368577207167651578553297938, −9.621875835566487876493018268320, −8.539866121027255912603647154679, −8.236451994783072162007565001063, −6.70050611261365736480099378246, −5.82678148160099584910215572988, −4.02709157117137369075997123664, −3.18479114723959621418810255129, −1.66136784998079231227713008756, −0.69570274335452160640611942480, 1.11591644617717784554228576472, 2.75980179333151237766430056604, 4.31195164346514827120201283176, 5.06349750084131179402539717566, 6.66340240983807495978096858029, 7.16229498861651554853591421634, 8.616489108274838051287344050981, 9.155231460146695749733615012799, 10.08322346189933902031858603593, 11.03567748981070995046271770462

Graph of the ZZ-function along the critical line