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Properties

Label 2-294-147.101-c1-0-8
Degree 22
Conductor 294294
Sign 0.803−0.595i0.803 - 0.595i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.930 − 0.365i)2-s + (1.16 + 1.27i)3-s + (0.733 + 0.680i)4-s + (3.54 + 2.41i)5-s + (−0.619 − 1.61i)6-s + (0.973 − 2.46i)7-s + (−0.433 − 0.900i)8-s + (−0.274 + 2.98i)9-s + (−2.41 − 3.54i)10-s + (−0.317 − 2.10i)11-s + (−0.0145 + 1.73i)12-s + (0.621 − 0.495i)13-s + (−1.80 + 1.93i)14-s + (1.04 + 7.36i)15-s + (0.0747 + 0.997i)16-s + (−7.16 − 2.21i)17-s + ⋯
L(s)  = 1  + (−0.658 − 0.258i)2-s + (0.673 + 0.738i)3-s + (0.366 + 0.340i)4-s + (1.58 + 1.08i)5-s + (−0.252 − 0.660i)6-s + (0.367 − 0.929i)7-s + (−0.153 − 0.318i)8-s + (−0.0914 + 0.995i)9-s + (−0.765 − 1.12i)10-s + (−0.0957 − 0.635i)11-s + (−0.00420 + 0.499i)12-s + (0.172 − 0.137i)13-s + (−0.482 + 0.516i)14-s + (0.270 + 1.90i)15-s + (0.0186 + 0.249i)16-s + (−1.73 − 0.536i)17-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=((0.803−0.595i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+1/2) L(s)=((0.803−0.595i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: 0.803−0.595i0.803 - 0.595i
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), 0.803−0.595i)(2,\ 294,\ (\ :1/2),\ 0.803 - 0.595i)

Particular Values

L(1)L(1) ≈\approx 1.40634+0.464275i1.40634 + 0.464275i
L(12)L(\frac12) ≈\approx 1.40634+0.464275i1.40634 + 0.464275i
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+(−1.16−1.27i)T 1 + (-1.16 - 1.27i)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}
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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

−11.18503654805816045423064166226, −10.67866918112596620955589717043, −10.11458304291874815538006317294, −9.215488363048619492217652102294, −8.326709011378899488246353198072, −7.04975820098991367910196917583, −6.13593890401865971227387019606, −4.53818737741043830303533127266, −3.04375963305569213371545345025, −2.06706684860733592834350772301, 1.66149007037297824206324079622, 2.30270609053053383247974156453, 4.80466382467439412508456889393, 5.96430201535233678258619535133, 6.72568843316875315989769972934, 8.194637591202660129393403660539, 8.950256022323884352356592573159, 9.277645556045365634853632368511, 10.43977014802321106803375220822, 11.87881387464725554613046192295

Graph of the ZZ-function along the critical line