↑

Properties

Label 2-294-147.104-c1-0-10
Degree 22
Conductor 294294
Sign 0.243+0.969i0.243 + 0.969i
Analytic cond. 2.347602.34760
Root an. cond. 1.532181.53218
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.781 − 0.623i)2-s + (−0.580 + 1.63i)3-s + (0.222 + 0.974i)4-s + (−3.14 − 1.51i)5-s + (1.47 − 0.913i)6-s + (2.47 − 0.926i)7-s + (0.433 − 0.900i)8-s + (−2.32 − 1.89i)9-s + (1.51 + 3.14i)10-s + (1.47 + 1.17i)11-s + (−1.72 − 0.202i)12-s + (−0.731 − 0.583i)13-s + (−2.51 − 0.820i)14-s + (4.30 − 4.25i)15-s + (−0.900 + 0.433i)16-s + (1.24 − 5.43i)17-s + ⋯
L(s)  = 1  + (−0.552 − 0.440i)2-s + (−0.335 + 0.942i)3-s + (0.111 + 0.487i)4-s + (−1.40 − 0.678i)5-s + (0.600 − 0.373i)6-s + (0.936 − 0.350i)7-s + (0.153 − 0.318i)8-s + (−0.775 − 0.631i)9-s + (0.479 + 0.996i)10-s + (0.445 + 0.355i)11-s + (−0.496 − 0.0585i)12-s + (−0.202 − 0.161i)13-s + (−0.672 − 0.219i)14-s + (1.11 − 1.09i)15-s + (−0.225 + 0.108i)16-s + (0.301 − 1.31i)17-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=((0.243+0.969i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+1/2) L(s)=((0.243+0.969i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: 0.243+0.969i0.243 + 0.969i
Analytic conductor: 2.347602.34760
Root analytic conductor: 1.532181.53218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ294(251,⋅)\chi_{294} (251, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 294, ( :1/2), 0.243+0.969i)(2,\ 294,\ (\ :1/2),\ 0.243 + 0.969i)

Particular Values

L(1)L(1) ≈\approx 0.498257−0.388685i0.498257 - 0.388685i
L(12)L(\frac12) ≈\approx 0.498257−0.388685i0.498257 - 0.388685i
L(32)L(\frac{3}{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+(0.781+0.623i)T 1 + (0.781 + 0.623i)T
3 1+(0.580−1.63i)T 1 + (0.580 - 1.63i)T
7 1+(−2.47+0.926i)T 1 + (-2.47 + 0.926i)T
good5 1+(3.14+1.51i)T+(3.11+3.90i)T2 1 + (3.14 + 1.51i)T + (3.11 + 3.90i)T^{2}
11 1+(−1.47−1.17i)T+(2.44+10.7i)T2 1 + (-1.47 - 1.17i)T + (2.44 + 10.7i)T^{2}
13 1+(0.731+0.583i)T+(2.89+12.6i)T2 1 + (0.731 + 0.583i)T + (2.89 + 12.6i)T^{2}
17 1+(−1.24+5.43i)T+(−15.3−7.37i)T2 1 + (-1.24 + 5.43i)T + (-15.3 - 7.37i)T^{2}
19 1+6.71iT−19T2 1 + 6.71iT - 19T^{2}
23 1+(−5.84+1.33i)T+(20.7−9.97i)T2 1 + (-5.84 + 1.33i)T + (20.7 - 9.97i)T^{2}
29 1+(−0.519−0.118i)T+(26.1+12.5i)T2 1 + (-0.519 - 0.118i)T + (26.1 + 12.5i)T^{2}
31 1+5.34iT−31T2 1 + 5.34iT - 31T^{2}
37 1+(−0.612+2.68i)T+(−33.3−16.0i)T2 1 + (-0.612 + 2.68i)T + (-33.3 - 16.0i)T^{2}
41 1+(1.61+0.777i)T+(25.5+32.0i)T2 1 + (1.61 + 0.777i)T + (25.5 + 32.0i)T^{2}
43 1+(9.41−4.53i)T+(26.8−33.6i)T2 1 + (9.41 - 4.53i)T + (26.8 - 33.6i)T^{2}
47 1+(7.08−8.88i)T+(−10.4−45.8i)T2 1 + (7.08 - 8.88i)T + (-10.4 - 45.8i)T^{2}
53 1+(5.46−1.24i)T+(47.7−22.9i)T2 1 + (5.46 - 1.24i)T + (47.7 - 22.9i)T^{2}
59 1+(−0.981+0.472i)T+(36.7−46.1i)T2 1 + (-0.981 + 0.472i)T + (36.7 - 46.1i)T^{2}
61 1+(−4.27−0.975i)T+(54.9+26.4i)T2 1 + (-4.27 - 0.975i)T + (54.9 + 26.4i)T^{2}
67 1+9.15T+67T2 1 + 9.15T + 67T^{2}
71 1+(−5.70+1.30i)T+(63.9−30.8i)T2 1 + (-5.70 + 1.30i)T + (63.9 - 30.8i)T^{2}
73 1+(−1.01+0.808i)T+(16.2−71.1i)T2 1 + (-1.01 + 0.808i)T + (16.2 - 71.1i)T^{2}
79 1−15.3T+79T2 1 - 15.3T + 79T^{2}
83 1+(9.92+12.4i)T+(−18.4+80.9i)T2 1 + (9.92 + 12.4i)T + (-18.4 + 80.9i)T^{2}
89 1+(6.59+8.26i)T+(−19.8+86.7i)T2 1 + (6.59 + 8.26i)T + (-19.8 + 86.7i)T^{2}
97 1−7.02iT−97T2 1 - 7.02iT - 97T^{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

−11.44947925990528168429847592386, −10.93242132426393762104400484456, −9.598919014796610654428323728251, −8.892247687688822880825720066494, −7.939643819978705402786835378509, −7.01579240093803431488456989334, −4.87730489534937937848323494401, −4.52602556208582595280854484064, −3.15639485500433124885669418187, −0.63894086524695306657206939858, 1.56383304850974682905679116151, 3.50769482688225769727492445708, 5.15278803020394189002166341833, 6.38886125704584341150272421176, 7.23662883521428122109256628088, 8.180306922487351239671132643403, 8.474585320008620934760324513639, 10.33905364253900924564896339794, 11.21137825767593250102309621648, 11.78484288559739730392795366238

Graph of the ZZ-function along the critical line