↑

Properties

Label 2-294-147.101-c1-0-17
Degree 22
Conductor 294294
Sign 0.547+0.836i0.547 + 0.836i
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  + (0.930 + 0.365i)2-s + (1.34 − 1.09i)3-s + (0.733 + 0.680i)4-s + (−3.54 − 2.41i)5-s + (1.65 − 0.524i)6-s + (0.973 − 2.46i)7-s + (0.433 + 0.900i)8-s + (0.618 − 2.93i)9-s + (−2.41 − 3.54i)10-s + (0.317 + 2.10i)11-s + (1.72 + 0.114i)12-s + (0.621 − 0.495i)13-s + (1.80 − 1.93i)14-s + (−7.41 + 0.618i)15-s + (0.0747 + 0.997i)16-s + (7.16 + 2.21i)17-s + ⋯
L(s)  = 1  + (0.658 + 0.258i)2-s + (0.776 − 0.630i)3-s + (0.366 + 0.340i)4-s + (−1.58 − 1.08i)5-s + (0.673 − 0.214i)6-s + (0.367 − 0.929i)7-s + (0.153 + 0.318i)8-s + (0.206 − 0.978i)9-s + (−0.765 − 1.12i)10-s + (0.0957 + 0.635i)11-s + (0.498 + 0.0331i)12-s + (0.172 − 0.137i)13-s + (0.482 − 0.516i)14-s + (−1.91 + 0.159i)15-s + (0.0186 + 0.249i)16-s + (1.73 + 0.536i)17-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=((0.547+0.836i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+1/2) L(s)=((0.547+0.836i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: 0.547+0.836i0.547 + 0.836i
Analytic conductor: 2.347602.34760
Root analytic conductor: 1.532181.53218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ294(101,⋅)\chi_{294} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 294, ( :1/2), 0.547+0.836i)(2,\ 294,\ (\ :1/2),\ 0.547 + 0.836i)

Particular Values

L(1)L(1) ≈\approx 1.70704−0.922672i1.70704 - 0.922672i
L(12)L(\frac12) ≈\approx 1.70704−0.922672i1.70704 - 0.922672i
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+(−0.930−0.365i)T 1 + (-0.930 - 0.365i)T
3 1+(−1.34+1.09i)T 1 + (-1.34 + 1.09i)T
7 1+(−0.973+2.46i)T 1 + (-0.973 + 2.46i)T
good5 1+(3.54+2.41i)T+(1.82+4.65i)T2 1 + (3.54 + 2.41i)T + (1.82 + 4.65i)T^{2}
11 1+(−0.317−2.10i)T+(−10.5+3.24i)T2 1 + (-0.317 - 2.10i)T + (-10.5 + 3.24i)T^{2}
13 1+(−0.621+0.495i)T+(2.89−12.6i)T2 1 + (-0.621 + 0.495i)T + (2.89 - 12.6i)T^{2}
17 1+(−7.16−2.21i)T+(14.0+9.57i)T2 1 + (-7.16 - 2.21i)T + (14.0 + 9.57i)T^{2}
19 1+(1.88−1.08i)T+(9.5−16.4i)T2 1 + (1.88 - 1.08i)T + (9.5 - 16.4i)T^{2}
23 1+(−1.36−4.43i)T+(−19.0+12.9i)T2 1 + (-1.36 - 4.43i)T + (-19.0 + 12.9i)T^{2}
29 1+(6.59−1.50i)T+(26.1−12.5i)T2 1 + (6.59 - 1.50i)T + (26.1 - 12.5i)T^{2}
31 1+(2.56+1.48i)T+(15.5+26.8i)T2 1 + (2.56 + 1.48i)T + (15.5 + 26.8i)T^{2}
37 1+(−0.202+0.187i)T+(2.76−36.8i)T2 1 + (-0.202 + 0.187i)T + (2.76 - 36.8i)T^{2}
41 1+(−5.29+2.54i)T+(25.5−32.0i)T2 1 + (-5.29 + 2.54i)T + (25.5 - 32.0i)T^{2}
43 1+(−4.05−1.95i)T+(26.8+33.6i)T2 1 + (-4.05 - 1.95i)T + (26.8 + 33.6i)T^{2}
47 1+(0.177−0.451i)T+(−34.4−31.9i)T2 1 + (0.177 - 0.451i)T + (-34.4 - 31.9i)T^{2}
53 1+(−3.48+3.75i)T+(−3.96−52.8i)T2 1 + (-3.48 + 3.75i)T + (-3.96 - 52.8i)T^{2}
59 1+(−3.31+2.26i)T+(21.5−54.9i)T2 1 + (-3.31 + 2.26i)T + (21.5 - 54.9i)T^{2}
61 1+(1.09+1.18i)T+(−4.55+60.8i)T2 1 + (1.09 + 1.18i)T + (-4.55 + 60.8i)T^{2}
67 1+(−2.16+3.75i)T+(−33.5−58.0i)T2 1 + (-2.16 + 3.75i)T + (-33.5 - 58.0i)T^{2}
71 1+(−2.36−0.540i)T+(63.9+30.8i)T2 1 + (-2.36 - 0.540i)T + (63.9 + 30.8i)T^{2}
73 1+(11.5−4.52i)T+(53.5−49.6i)T2 1 + (11.5 - 4.52i)T + (53.5 - 49.6i)T^{2}
79 1+(−5.04−8.74i)T+(−39.5+68.4i)T2 1 + (-5.04 - 8.74i)T + (-39.5 + 68.4i)T^{2}
83 1+(7.49−9.39i)T+(−18.4−80.9i)T2 1 + (7.49 - 9.39i)T + (-18.4 - 80.9i)T^{2}
89 1+(5.32+0.803i)T+(85.0+26.2i)T2 1 + (5.32 + 0.803i)T + (85.0 + 26.2i)T^{2}
97 1−3.52iT−97T2 1 - 3.52iT - 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

−11.99927249761368912518313432871, −11.04937872703631124363019498454, −9.517533479811425931192947541025, −8.300705014421327723272632424245, −7.68828393082573082865471406920, −7.18918362379481073128004045646, −5.42412472650619859643802655989, −4.07123723101514385114716364575, −3.60710530435804225854873142675, −1.30048789037884062168529084630, 2.68060236393400193227796475766, 3.43815802724841039355176605681, 4.41960179528052652508697074232, 5.75108046102807883747322442404, 7.25631127676510160922030518829, 8.032762455889741229730903510258, 9.003788858882179921858684371134, 10.34925610715215602424444066255, 11.13707415528664252483874027891, 11.76046521481850453274485202722

Graph of the ZZ-function along the critical line