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Properties

Label 2-294-147.101-c1-0-0
Degree 22
Conductor 294294
Sign −0.364−0.931i-0.364 - 0.931i
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 + (−0.0758 − 1.73i)3-s + (0.733 + 0.680i)4-s + (−2.77 − 1.89i)5-s + (−0.561 + 1.63i)6-s + (−2.14 + 1.54i)7-s + (−0.433 − 0.900i)8-s + (−2.98 + 0.262i)9-s + (1.89 + 2.77i)10-s + (0.905 + 6.00i)11-s + (1.12 − 1.32i)12-s + (1.32 − 1.05i)13-s + (2.56 − 0.654i)14-s + (−3.06 + 4.94i)15-s + (0.0747 + 0.997i)16-s + (−0.409 − 0.126i)17-s + ⋯
L(s)  = 1  + (−0.658 − 0.258i)2-s + (−0.0438 − 0.999i)3-s + (0.366 + 0.340i)4-s + (−1.24 − 0.845i)5-s + (−0.229 + 0.668i)6-s + (−0.811 + 0.584i)7-s + (−0.153 − 0.318i)8-s + (−0.996 + 0.0875i)9-s + (0.597 + 0.876i)10-s + (0.272 + 1.81i)11-s + (0.323 − 0.381i)12-s + (0.368 − 0.293i)13-s + (0.685 − 0.174i)14-s + (−0.790 + 1.27i)15-s + (0.0186 + 0.249i)16-s + (−0.0993 − 0.0306i)17-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=((−0.364−0.931i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+1/2) L(s)=((−0.364−0.931i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: −0.364−0.931i-0.364 - 0.931i
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), −0.364−0.931i)(2,\ 294,\ (\ :1/2),\ -0.364 - 0.931i)

Particular Values

L(1)L(1) ≈\approx 0.0117487+0.0172140i0.0117487 + 0.0172140i
L(12)L(\frac12) ≈\approx 0.0117487+0.0172140i0.0117487 + 0.0172140i
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+(0.0758+1.73i)T 1 + (0.0758 + 1.73i)T
7 1+(2.14−1.54i)T 1 + (2.14 - 1.54i)T
good5 1+(2.77+1.89i)T+(1.82+4.65i)T2 1 + (2.77 + 1.89i)T + (1.82 + 4.65i)T^{2}
11 1+(−0.905−6.00i)T+(−10.5+3.24i)T2 1 + (-0.905 - 6.00i)T + (-10.5 + 3.24i)T^{2}
13 1+(−1.32+1.05i)T+(2.89−12.6i)T2 1 + (-1.32 + 1.05i)T + (2.89 - 12.6i)T^{2}
17 1+(0.409+0.126i)T+(14.0+9.57i)T2 1 + (0.409 + 0.126i)T + (14.0 + 9.57i)T^{2}
19 1+(−0.985+0.568i)T+(9.5−16.4i)T2 1 + (-0.985 + 0.568i)T + (9.5 - 16.4i)T^{2}
23 1+(1.51+4.91i)T+(−19.0+12.9i)T2 1 + (1.51 + 4.91i)T + (-19.0 + 12.9i)T^{2}
29 1+(7.31−1.66i)T+(26.1−12.5i)T2 1 + (7.31 - 1.66i)T + (26.1 - 12.5i)T^{2}
31 1+(1.31+0.757i)T+(15.5+26.8i)T2 1 + (1.31 + 0.757i)T + (15.5 + 26.8i)T^{2}
37 1+(7.65−7.09i)T+(2.76−36.8i)T2 1 + (7.65 - 7.09i)T + (2.76 - 36.8i)T^{2}
41 1+(9.57−4.61i)T+(25.5−32.0i)T2 1 + (9.57 - 4.61i)T + (25.5 - 32.0i)T^{2}
43 1+(6.51+3.13i)T+(26.8+33.6i)T2 1 + (6.51 + 3.13i)T + (26.8 + 33.6i)T^{2}
47 1+(−1.97+5.02i)T+(−34.4−31.9i)T2 1 + (-1.97 + 5.02i)T + (-34.4 - 31.9i)T^{2}
53 1+(0.516−0.556i)T+(−3.96−52.8i)T2 1 + (0.516 - 0.556i)T + (-3.96 - 52.8i)T^{2}
59 1+(7.56−5.15i)T+(21.5−54.9i)T2 1 + (7.56 - 5.15i)T + (21.5 - 54.9i)T^{2}
61 1+(3.47+3.74i)T+(−4.55+60.8i)T2 1 + (3.47 + 3.74i)T + (-4.55 + 60.8i)T^{2}
67 1+(−3.86+6.69i)T+(−33.5−58.0i)T2 1 + (-3.86 + 6.69i)T + (-33.5 - 58.0i)T^{2}
71 1+(−13.2−3.03i)T+(63.9+30.8i)T2 1 + (-13.2 - 3.03i)T + (63.9 + 30.8i)T^{2}
73 1+(1.42−0.559i)T+(53.5−49.6i)T2 1 + (1.42 - 0.559i)T + (53.5 - 49.6i)T^{2}
79 1+(2.37+4.12i)T+(−39.5+68.4i)T2 1 + (2.37 + 4.12i)T + (-39.5 + 68.4i)T^{2}
83 1+(−2.83+3.54i)T+(−18.4−80.9i)T2 1 + (-2.83 + 3.54i)T + (-18.4 - 80.9i)T^{2}
89 1+(−3.81−0.574i)T+(85.0+26.2i)T2 1 + (-3.81 - 0.574i)T + (85.0 + 26.2i)T^{2}
97 1+4.18iT−97T2 1 + 4.18iT - 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

−12.26528411824243717030465250523, −11.45612666168808340721920010101, −10.08133075387657649237021637199, −8.998020287591067928581220165680, −8.300598139873502154975543737521, −7.34815201221961675003617173696, −6.57836011980463426679776374717, −4.98093301312818178153865369674, −3.46224866567673495702075719463, −1.84123487052234960793802802321, 0.01855470263861703250851748164, 3.41378309069636032261004374215, 3.69453475354283410941767010358, 5.64316236714076041051691402235, 6.66375163640761394731865847999, 7.73198353223053309667435484078, 8.691095930991834223908704469060, 9.585274821953043280062127311816, 10.72667526650137426590154410351, 11.10252850563848246108792181254

Graph of the ZZ-function along the critical line