↑

Properties

Label 2-294-7.4-c5-0-8
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 3.46i)2-s + (−4.5 + 7.79i)3-s + (−7.99 + 13.8i)4-s + (43 + 74.4i)5-s − 36·6-s − 63.9·8-s + (−40.5 − 70.1i)9-s + (−172 + 297. i)10-s + (−17 + 29.4i)11-s + (−72 − 124. i)12-s + 3·13-s − 774.·15-s + (−128 − 221. i)16-s + (−952 + 1.64e3i)17-s + (162 − 280. i)18-s + (−744.5 − 1.28e3i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.769 + 1.33i)5-s − 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.543 + 0.942i)10-s + (−0.0423 + 0.0733i)11-s + (−0.144 − 0.249i)12-s + 0.00492·13-s − 0.888·15-s + (−0.125 − 0.216i)16-s + (−0.798 + 1.38i)17-s + (0.117 − 0.204i)18-s + (−0.473 − 0.819i)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 1.1692852611.169285261
L(12)L(\frac12) ≈\approx 1.1692852611.169285261
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+(−43−74.4i)T+(−1.56e3+2.70e3i)T2 1 + (-43 - 74.4i)T + (-1.56e3 + 2.70e3i)T^{2}
11 1+(17−29.4i)T+(−8.05e4−1.39e5i)T2 1 + (17 - 29.4i)T + (-8.05e4 - 1.39e5i)T^{2}
13 1−3T+3.71e5T2 1 - 3T + 3.71e5T^{2}
17 1+(952−1.64e3i)T+(−7.09e5−1.22e6i)T2 1 + (952 - 1.64e3i)T + (-7.09e5 - 1.22e6i)T^{2}
19 1+(744.5+1.28e3i)T+(−1.23e6+2.14e6i)T2 1 + (744.5 + 1.28e3i)T + (-1.23e6 + 2.14e6i)T^{2}
23 1+(−112−193.i)T+(−3.21e6+5.57e6i)T2 1 + (-112 - 193. i)T + (-3.21e6 + 5.57e6i)T^{2}
29 1+6.50e3T+2.05e7T2 1 + 6.50e3T + 2.05e7T^{2}
31 1+(−865.5+1.49e3i)T+(−1.43e7−2.47e7i)T2 1 + (-865.5 + 1.49e3i)T + (-1.43e7 - 2.47e7i)T^{2}
37 1+(−3.81e3−6.61e3i)T+(−3.46e7+6.00e7i)T2 1 + (-3.81e3 - 6.61e3i)T + (-3.46e7 + 6.00e7i)T^{2}
41 1+1.54e4T+1.15e8T2 1 + 1.54e4T + 1.15e8T^{2}
43 1−1.84e4T+1.47e8T2 1 - 1.84e4T + 1.47e8T^{2}
47 1+(−9.23e3−1.59e4i)T+(−1.14e8+1.98e8i)T2 1 + (-9.23e3 - 1.59e4i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(−9.97e3+1.72e4i)T+(−2.09e8−3.62e8i)T2 1 + (-9.97e3 + 1.72e4i)T + (-2.09e8 - 3.62e8i)T^{2}
59 1+(1.59e4−2.75e4i)T+(−3.57e8−6.19e8i)T2 1 + (1.59e4 - 2.75e4i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(2.88e4+4.99e4i)T+(−4.22e8+7.31e8i)T2 1 + (2.88e4 + 4.99e4i)T + (-4.22e8 + 7.31e8i)T^{2}
67 1+(−3.02e4+5.24e4i)T+(−6.75e8−1.16e9i)T2 1 + (-3.02e4 + 5.24e4i)T + (-6.75e8 - 1.16e9i)T^{2}
71 1+4.48e4T+1.80e9T2 1 + 4.48e4T + 1.80e9T^{2}
73 1+(−1.04e4+1.80e4i)T+(−1.03e9−1.79e9i)T2 1 + (-1.04e4 + 1.80e4i)T + (-1.03e9 - 1.79e9i)T^{2}
79 1+(−1.52e4−2.64e4i)T+(−1.53e9+2.66e9i)T2 1 + (-1.52e4 - 2.64e4i)T + (-1.53e9 + 2.66e9i)T^{2}
83 1+1.10e5T+3.93e9T2 1 + 1.10e5T + 3.93e9T^{2}
89 1+(2.94e4+5.10e4i)T+(−2.79e9+4.83e9i)T2 1 + (2.94e4 + 5.10e4i)T + (-2.79e9 + 4.83e9i)T^{2}
97 1−1.19e5T+8.58e9T2 1 - 1.19e5T + 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

−11.27387736229650752234088444444, −10.71931116464294747510896777831, −9.768045403887808591763810122435, −8.774284993239319606535949538733, −7.44498123599887760610706494109, −6.43624624542158109095244120233, −5.91022029342992991163260945346, −4.56765105067927125306035083518, −3.39851478785159545803861195973, −2.14641973567871489123447945075, 0.27647657844965387612131433877, 1.40329686489076269739106779520, 2.41966581545510576503914260785, 4.19065861651640633201959575238, 5.23193735000298371912898166415, 5.93230521568187760191337984999, 7.29495143853881468371535156304, 8.682455427172272337749185291269, 9.306327293045087413871770529446, 10.35552324586099047025922338445

Graph of the ZZ-function along the critical line