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Properties

Label 2-294-7.2-c5-0-20
Degree 22
Conductor 294294
Sign 0.386+0.922i0.386 + 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 + (11.2 − 19.4i)5-s + 36·6-s + 63.9·8-s + (−40.5 + 70.1i)9-s + (44.9 + 77.9i)10-s + (−170. − 294. i)11-s + (−72 + 124. i)12-s + 728.·13-s − 202.·15-s + (−128 + 221. i)16-s + (404. + 701. i)17-s + (−162 − 280. i)18-s + (513. − 888. i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.201 − 0.348i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.142 + 0.246i)10-s + (−0.424 − 0.734i)11-s + (−0.144 + 0.249i)12-s + 1.19·13-s − 0.232·15-s + (−0.125 + 0.216i)16-s + (0.339 + 0.588i)17-s + (−0.117 − 0.204i)18-s + (0.326 − 0.564i)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+0.922i0.386 + 0.922i
Analytic conductor: 47.152847.1528
Root analytic conductor: 6.866796.86679
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ294(79,⋅)\chi_{294} (79, \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 1.3289051471.328905147
L(12)L(\frac12) ≈\approx 1.3289051471.328905147
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+(−11.2+19.4i)T+(−1.56e3−2.70e3i)T2 1 + (-11.2 + 19.4i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(170.+294.i)T+(−8.05e4+1.39e5i)T2 1 + (170. + 294. i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1−728.T+3.71e5T2 1 - 728.T + 3.71e5T^{2}
17 1+(−404.−701.i)T+(−7.09e5+1.22e6i)T2 1 + (-404. - 701. i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(−513.+888.i)T+(−1.23e6−2.14e6i)T2 1 + (-513. + 888. i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(711.−1.23e3i)T+(−3.21e6−5.57e6i)T2 1 + (711. - 1.23e3i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1−5.21e3T+2.05e7T2 1 - 5.21e3T + 2.05e7T^{2}
31 1+(3.51e3+6.09e3i)T+(−1.43e7+2.47e7i)T2 1 + (3.51e3 + 6.09e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(6.39e3−1.10e4i)T+(−3.46e7−6.00e7i)T2 1 + (6.39e3 - 1.10e4i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1+1.17e3T+1.15e8T2 1 + 1.17e3T + 1.15e8T^{2}
43 1−3.66e3T+1.47e8T2 1 - 3.66e3T + 1.47e8T^{2}
47 1+(−4.65e3+8.06e3i)T+(−1.14e8−1.98e8i)T2 1 + (-4.65e3 + 8.06e3i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(1.78e4+3.08e4i)T+(−2.09e8+3.62e8i)T2 1 + (1.78e4 + 3.08e4i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(1.51e4+2.63e4i)T+(−3.57e8+6.19e8i)T2 1 + (1.51e4 + 2.63e4i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(−1.60e4+2.78e4i)T+(−4.22e8−7.31e8i)T2 1 + (-1.60e4 + 2.78e4i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(1.06e4+1.84e4i)T+(−6.75e8+1.16e9i)T2 1 + (1.06e4 + 1.84e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1−6.11e4T+1.80e9T2 1 - 6.11e4T + 1.80e9T^{2}
73 1+(−2.06e4−3.57e4i)T+(−1.03e9+1.79e9i)T2 1 + (-2.06e4 - 3.57e4i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(−1.75e4+3.03e4i)T+(−1.53e9−2.66e9i)T2 1 + (-1.75e4 + 3.03e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1+8.61e4T+3.93e9T2 1 + 8.61e4T + 3.93e9T^{2}
89 1+(−3.89e4+6.75e4i)T+(−2.79e9−4.83e9i)T2 1 + (-3.89e4 + 6.75e4i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1−1.61e5T+8.58e9T2 1 - 1.61e5T + 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

−10.80027010386463807244771222791, −9.678637711320721846694431433760, −8.579720729718251855214782708841, −7.996093207311133050971425787702, −6.75756850157364672983408263036, −5.91126277404953145765458783994, −5.04821341870029481448958778208, −3.42034549365550281877376625769, −1.60485374640931317454032050579, −0.50703506543731530270446824941, 1.08414002896829867392297372945, 2.57326640511419660239006376063, 3.73973190031628130907894059642, 4.89996542908997233300932179086, 6.11807911051165345421185958271, 7.31594869983354040218129889810, 8.492095190300712325178801568986, 9.391034478897736813604934927572, 10.46898081972947847047049845440, 10.71673538095802339354820289310

Graph of the ZZ-function along the critical line