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Properties

Label 2-294-147.101-c1-0-3
Degree 22
Conductor 294294
Sign 0.384−0.922i0.384 - 0.922i
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 + (−1.67 − 0.425i)3-s + (0.733 + 0.680i)4-s + (0.562 + 0.383i)5-s + (−1.40 − 1.00i)6-s + (−1.02 + 2.43i)7-s + (0.433 + 0.900i)8-s + (2.63 + 1.42i)9-s + (0.383 + 0.562i)10-s + (0.601 + 3.99i)11-s + (−0.941 − 1.45i)12-s + (1.92 − 1.53i)13-s + (−1.84 + 1.89i)14-s + (−0.780 − 0.883i)15-s + (0.0747 + 0.997i)16-s + (1.73 + 0.533i)17-s + ⋯
L(s)  = 1  + (0.658 + 0.258i)2-s + (−0.969 − 0.245i)3-s + (0.366 + 0.340i)4-s + (0.251 + 0.171i)5-s + (−0.574 − 0.412i)6-s + (−0.388 + 0.921i)7-s + (0.153 + 0.318i)8-s + (0.879 + 0.476i)9-s + (0.121 + 0.177i)10-s + (0.181 + 1.20i)11-s + (−0.271 − 0.419i)12-s + (0.534 − 0.426i)13-s + (−0.493 + 0.505i)14-s + (−0.201 − 0.227i)15-s + (0.0186 + 0.249i)16-s + (0.419 + 0.129i)17-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=((0.384−0.922i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+1/2) L(s)=((0.384−0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: 0.384−0.922i0.384 - 0.922i
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), 0.384−0.922i)(2,\ 294,\ (\ :1/2),\ 0.384 - 0.922i)

Particular Values

L(1)L(1) ≈\approx 1.17417+0.782466i1.17417 + 0.782466i
L(12)L(\frac12) ≈\approx 1.17417+0.782466i1.17417 + 0.782466i
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+(1.67+0.425i)T 1 + (1.67 + 0.425i)T
7 1+(1.02−2.43i)T 1 + (1.02 - 2.43i)T
good5 1+(−0.562−0.383i)T+(1.82+4.65i)T2 1 + (-0.562 - 0.383i)T + (1.82 + 4.65i)T^{2}
11 1+(−0.601−3.99i)T+(−10.5+3.24i)T2 1 + (-0.601 - 3.99i)T + (-10.5 + 3.24i)T^{2}
13 1+(−1.92+1.53i)T+(2.89−12.6i)T2 1 + (-1.92 + 1.53i)T + (2.89 - 12.6i)T^{2}
17 1+(−1.73−0.533i)T+(14.0+9.57i)T2 1 + (-1.73 - 0.533i)T + (14.0 + 9.57i)T^{2}
19 1+(−1.85+1.07i)T+(9.5−16.4i)T2 1 + (-1.85 + 1.07i)T + (9.5 - 16.4i)T^{2}
23 1+(−2.52−8.18i)T+(−19.0+12.9i)T2 1 + (-2.52 - 8.18i)T + (-19.0 + 12.9i)T^{2}
29 1+(5.96−1.36i)T+(26.1−12.5i)T2 1 + (5.96 - 1.36i)T + (26.1 - 12.5i)T^{2}
31 1+(4.37+2.52i)T+(15.5+26.8i)T2 1 + (4.37 + 2.52i)T + (15.5 + 26.8i)T^{2}
37 1+(−6.91+6.41i)T+(2.76−36.8i)T2 1 + (-6.91 + 6.41i)T + (2.76 - 36.8i)T^{2}
41 1+(6.53−3.14i)T+(25.5−32.0i)T2 1 + (6.53 - 3.14i)T + (25.5 - 32.0i)T^{2}
43 1+(−1.51−0.731i)T+(26.8+33.6i)T2 1 + (-1.51 - 0.731i)T + (26.8 + 33.6i)T^{2}
47 1+(−4.27+10.8i)T+(−34.4−31.9i)T2 1 + (-4.27 + 10.8i)T + (-34.4 - 31.9i)T^{2}
53 1+(−7.86+8.47i)T+(−3.96−52.8i)T2 1 + (-7.86 + 8.47i)T + (-3.96 - 52.8i)T^{2}
59 1+(0.00789−0.00538i)T+(21.5−54.9i)T2 1 + (0.00789 - 0.00538i)T + (21.5 - 54.9i)T^{2}
61 1+(−2.82−3.04i)T+(−4.55+60.8i)T2 1 + (-2.82 - 3.04i)T + (-4.55 + 60.8i)T^{2}
67 1+(−3.82+6.62i)T+(−33.5−58.0i)T2 1 + (-3.82 + 6.62i)T + (-33.5 - 58.0i)T^{2}
71 1+(5.04+1.15i)T+(63.9+30.8i)T2 1 + (5.04 + 1.15i)T + (63.9 + 30.8i)T^{2}
73 1+(4.71−1.84i)T+(53.5−49.6i)T2 1 + (4.71 - 1.84i)T + (53.5 - 49.6i)T^{2}
79 1+(−1.53−2.65i)T+(−39.5+68.4i)T2 1 + (-1.53 - 2.65i)T + (-39.5 + 68.4i)T^{2}
83 1+(0.403−0.505i)T+(−18.4−80.9i)T2 1 + (0.403 - 0.505i)T + (-18.4 - 80.9i)T^{2}
89 1+(−10.5−1.59i)T+(85.0+26.2i)T2 1 + (-10.5 - 1.59i)T + (85.0 + 26.2i)T^{2}
97 1+9.68iT−97T2 1 + 9.68iT - 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

−11.96890534635373256602670511554, −11.38463242907373489026420556351, −10.16857327385646447136464580625, −9.286164932601183087197944496426, −7.70483654700955826162640565295, −6.87163777401084216727494327779, −5.78122358846713221941574844140, −5.25355712642200064683113664614, −3.75283755323079835062253397239, −2.02486784880594905487907834369, 1.06072911517188429279514998942, 3.39294845618648860709279715111, 4.36964701194552636227148752528, 5.60621022619479992522105035910, 6.34812293308006832940445187547, 7.37504403240431869121198234923, 8.998751911147031036432994566886, 10.05918447134685454506299825295, 10.90411695222140281189244283674, 11.45299949231272188123741086494

Graph of the ZZ-function along the critical line