↑

Properties

Label 2-294-7.2-c5-0-5
Degree 22
Conductor 294294
Sign −0.900+0.435i-0.900 + 0.435i
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 + (−23.4 + 40.6i)5-s − 36·6-s + 63.9·8-s + (−40.5 + 70.1i)9-s + (−93.8 − 162. i)10-s + (43.7 + 75.7i)11-s + (72 − 124. i)12-s + 754.·13-s − 422.·15-s + (−128 + 221. i)16-s + (724. + 1.25e3i)17-s + (−162 − 280. i)18-s + (−1.27e3 + 2.19e3i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.419 + 0.727i)5-s − 0.408·6-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.296 − 0.514i)10-s + (0.108 + 0.188i)11-s + (0.144 − 0.249i)12-s + 1.23·13-s − 0.484·15-s + (−0.125 + 0.216i)16-s + (0.608 + 1.05i)17-s + (−0.117 − 0.204i)18-s + (−0.807 + 1.39i)19-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=((−0.900+0.435i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+5/2) L(s)=((−0.900+0.435i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: −0.900+0.435i-0.900 + 0.435i
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), −0.900+0.435i)(2,\ 294,\ (\ :5/2),\ -0.900 + 0.435i)

Particular Values

L(3)L(3) ≈\approx 1.1761552741.176155274
L(12)L(\frac12) ≈\approx 1.1761552741.176155274
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+(23.4−40.6i)T+(−1.56e3−2.70e3i)T2 1 + (23.4 - 40.6i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(−43.7−75.7i)T+(−8.05e4+1.39e5i)T2 1 + (-43.7 - 75.7i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1−754.T+3.71e5T2 1 - 754.T + 3.71e5T^{2}
17 1+(−724.−1.25e3i)T+(−7.09e5+1.22e6i)T2 1 + (-724. - 1.25e3i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(1.27e3−2.19e3i)T+(−1.23e6−2.14e6i)T2 1 + (1.27e3 - 2.19e3i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(−456.+790.i)T+(−3.21e6−5.57e6i)T2 1 + (-456. + 790. i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1−173.T+2.05e7T2 1 - 173.T + 2.05e7T^{2}
31 1+(2.26e3+3.92e3i)T+(−1.43e7+2.47e7i)T2 1 + (2.26e3 + 3.92e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(3.41e3−5.91e3i)T+(−3.46e7−6.00e7i)T2 1 + (3.41e3 - 5.91e3i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1−1.30e4T+1.15e8T2 1 - 1.30e4T + 1.15e8T^{2}
43 1+1.22e4T+1.47e8T2 1 + 1.22e4T + 1.47e8T^{2}
47 1+(6.74e3−1.16e4i)T+(−1.14e8−1.98e8i)T2 1 + (6.74e3 - 1.16e4i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(−4.83e3−8.38e3i)T+(−2.09e8+3.62e8i)T2 1 + (-4.83e3 - 8.38e3i)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+(366.−634.i)T+(−4.22e8−7.31e8i)T2 1 + (366. - 634. i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(2.31e4+4.01e4i)T+(−6.75e8+1.16e9i)T2 1 + (2.31e4 + 4.01e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1+3.29e3T+1.80e9T2 1 + 3.29e3T + 1.80e9T^{2}
73 1+(5.11e3+8.86e3i)T+(−1.03e9+1.79e9i)T2 1 + (5.11e3 + 8.86e3i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(4.92e4−8.53e4i)T+(−1.53e9−2.66e9i)T2 1 + (4.92e4 - 8.53e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1−8.77e4T+3.93e9T2 1 - 8.77e4T + 3.93e9T^{2}
89 1+(−3.45e4+5.98e4i)T+(−2.79e9−4.83e9i)T2 1 + (-3.45e4 + 5.98e4i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1+4.21e4T+8.58e9T2 1 + 4.21e4T + 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

−11.04115310106611554450762195435, −10.53973989868688430335925150393, −9.544882874828582922254853343161, −8.414651356523668557045321119465, −7.84614529250371283784733395733, −6.56056129932081151486680461647, −5.76308750418008092983988042824, −4.20965514256745302725831824676, −3.35204625488756526396846502376, −1.56522024529313494043461495039, 0.37562797687600049170435455348, 1.30819071922384133055059544166, 2.76920660568826199405912575345, 3.93860864431990098658573006785, 5.16724962866889986672325826905, 6.64420643492239765187532363590, 7.71361956170166360376330762218, 8.741863026160127868231019191561, 9.104546410430042097618282233835, 10.52569008632735801476241808497

Graph of the ZZ-function along the critical line