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Properties

Label 2-294-21.20-c5-0-16
Degree 22
Conductor 294294
Sign −0.881+0.472i-0.881 + 0.472i
Analytic cond. 47.152847.1528
Root an. cond. 6.866796.86679
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s + (12.3 + 9.52i)3-s − 16·4-s − 12.2·5-s + (−38.1 + 49.3i)6-s − 64i·8-s + (61.5 + 235. i)9-s − 48.8i·10-s + 198. i·11-s + (−197. − 152. i)12-s + 369. i·13-s + (−150. − 116. i)15-s + 256·16-s + 896.·17-s + (−940. + 246. i)18-s + 1.56e3i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.791 + 0.611i)3-s − 0.5·4-s − 0.218·5-s + (−0.432 + 0.559i)6-s − 0.353i·8-s + (0.253 + 0.967i)9-s − 0.154i·10-s + 0.494i·11-s + (−0.395 − 0.305i)12-s + 0.606i·13-s + (−0.172 − 0.133i)15-s + 0.250·16-s + 0.752·17-s + (−0.684 + 0.179i)18-s + 0.993i·19-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=((−0.881+0.472i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+5/2) L(s)=((−0.881+0.472i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: −0.881+0.472i-0.881 + 0.472i
Analytic conductor: 47.152847.1528
Root analytic conductor: 6.866796.86679
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ294(293,⋅)\chi_{294} (293, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 294, ( :5/2), −0.881+0.472i)(2,\ 294,\ (\ :5/2),\ -0.881 + 0.472i)

Particular Values

L(3)L(3) ≈\approx 1.5074693001.507469300
L(12)L(\frac12) ≈\approx 1.5074693001.507469300
L(72)L(\frac{7}{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 1−4iT 1 - 4iT
3 1+(−12.3−9.52i)T 1 + (-12.3 - 9.52i)T
7 1 1
good5 1+12.2T+3.12e3T2 1 + 12.2T + 3.12e3T^{2}
11 1−198.iT−1.61e5T2 1 - 198. iT - 1.61e5T^{2}
13 1−369.iT−3.71e5T2 1 - 369. iT - 3.71e5T^{2}
17 1−896.T+1.41e6T2 1 - 896.T + 1.41e6T^{2}
19 1−1.56e3iT−2.47e6T2 1 - 1.56e3iT - 2.47e6T^{2}
23 1+793.iT−6.43e6T2 1 + 793. iT - 6.43e6T^{2}
29 1−496.iT−2.05e7T2 1 - 496. iT - 2.05e7T^{2}
31 1+2.72e3iT−2.86e7T2 1 + 2.72e3iT - 2.86e7T^{2}
37 1−6.46e3T+6.93e7T2 1 - 6.46e3T + 6.93e7T^{2}
41 1+2.07e4T+1.15e8T2 1 + 2.07e4T + 1.15e8T^{2}
43 1+2.05e4T+1.47e8T2 1 + 2.05e4T + 1.47e8T^{2}
47 1+8.20e3T+2.29e8T2 1 + 8.20e3T + 2.29e8T^{2}
53 1−1.69e4iT−4.18e8T2 1 - 1.69e4iT - 4.18e8T^{2}
59 1+6.61e3T+7.14e8T2 1 + 6.61e3T + 7.14e8T^{2}
61 1+5.55e4iT−8.44e8T2 1 + 5.55e4iT - 8.44e8T^{2}
67 1−2.57e4T+1.35e9T2 1 - 2.57e4T + 1.35e9T^{2}
71 1−7.44e4iT−1.80e9T2 1 - 7.44e4iT - 1.80e9T^{2}
73 1−3.30e4iT−2.07e9T2 1 - 3.30e4iT - 2.07e9T^{2}
79 1−7.26e4T+3.07e9T2 1 - 7.26e4T + 3.07e9T^{2}
83 1+4.70e4T+3.93e9T2 1 + 4.70e4T + 3.93e9T^{2}
89 1+1.19e4T+5.58e9T2 1 + 1.19e4T + 5.58e9T^{2}
97 1+8.48e4iT−8.58e9T2 1 + 8.48e4iT - 8.58e9T^{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

−11.44618693392962602785638540054, −10.06392869173096888858793719138, −9.646074494639052929727728255564, −8.418556204769572513639815656361, −7.83894332520388345279317265610, −6.73269375603385134327710905109, −5.40183721022034511777578469452, −4.33572359669992332368494719851, −3.40460250498688053615316063612, −1.79093661003629559881893998920, 0.35614375049210587406643120583, 1.57801649151986193240858033971, 2.90208410644541028529495577971, 3.68117666176465554429877731445, 5.18668241627423764910503638508, 6.55558640530163941385596390713, 7.76253641320282931167160896033, 8.453227673955830409935517620819, 9.455065331623830950572725998693, 10.32845332942050320318111892741

Graph of the ZZ-function along the critical line