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Properties

Label 2-14e2-28.27-c5-0-30
Degree 22
Conductor 196196
Sign 0.997+0.0737i0.997 + 0.0737i
Analytic cond. 31.435231.4352
Root an. cond. 5.606715.60671
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  + (−5.48 − 1.38i)2-s − 26.5·3-s + (28.1 + 15.2i)4-s + 52.9i·5-s + (145. + 36.7i)6-s + (−133. − 122. i)8-s + 460.·9-s + (73.4 − 290. i)10-s + 105. i·11-s + (−746. − 403. i)12-s − 827. i·13-s − 1.40e3i·15-s + (561. + 856. i)16-s + 1.75e3i·17-s + (−2.52e3 − 638. i)18-s − 3.02e3·19-s + ⋯
L(s)  = 1  + (−0.969 − 0.245i)2-s − 1.70·3-s + (0.879 + 0.475i)4-s + 0.947i·5-s + (1.64 + 0.417i)6-s + (−0.736 − 0.676i)8-s + 1.89·9-s + (0.232 − 0.919i)10-s + 0.263i·11-s + (−1.49 − 0.808i)12-s − 1.35i·13-s − 1.61i·15-s + (0.548 + 0.836i)16-s + 1.47i·17-s + (−1.83 − 0.464i)18-s − 1.92·19-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((0.997+0.0737i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0737i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+5/2) L(s)=((0.997+0.0737i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0737i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: 0.997+0.0737i0.997 + 0.0737i
Analytic conductor: 31.435231.4352
Root analytic conductor: 5.606715.60671
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ196(195,⋅)\chi_{196} (195, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 196, ( :5/2), 0.997+0.0737i)(2,\ 196,\ (\ :5/2),\ 0.997 + 0.0737i)

Particular Values

L(3)L(3) ≈\approx 0.39218988520.3921898852
L(12)L(\frac12) ≈\approx 0.39218988520.3921898852
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+(5.48+1.38i)T 1 + (5.48 + 1.38i)T
7 1 1
good3 1+26.5T+243T2 1 + 26.5T + 243T^{2}
5 1−52.9iT−3.12e3T2 1 - 52.9iT - 3.12e3T^{2}
11 1−105.iT−1.61e5T2 1 - 105. iT - 1.61e5T^{2}
13 1+827.iT−3.71e5T2 1 + 827. iT - 3.71e5T^{2}
17 1−1.75e3iT−1.41e6T2 1 - 1.75e3iT - 1.41e6T^{2}
19 1+3.02e3T+2.47e6T2 1 + 3.02e3T + 2.47e6T^{2}
23 1+2.13e3iT−6.43e6T2 1 + 2.13e3iT - 6.43e6T^{2}
29 1+3.80e3T+2.05e7T2 1 + 3.80e3T + 2.05e7T^{2}
31 1+7.10e3T+2.86e7T2 1 + 7.10e3T + 2.86e7T^{2}
37 1+202.T+6.93e7T2 1 + 202.T + 6.93e7T^{2}
41 1+7.21e3iT−1.15e8T2 1 + 7.21e3iT - 1.15e8T^{2}
43 1+1.39e4iT−1.47e8T2 1 + 1.39e4iT - 1.47e8T^{2}
47 1−1.33e4T+2.29e8T2 1 - 1.33e4T + 2.29e8T^{2}
53 1−1.19e4T+4.18e8T2 1 - 1.19e4T + 4.18e8T^{2}
59 1−1.30e3T+7.14e8T2 1 - 1.30e3T + 7.14e8T^{2}
61 1−1.35e4iT−8.44e8T2 1 - 1.35e4iT - 8.44e8T^{2}
67 1+3.99e4iT−1.35e9T2 1 + 3.99e4iT - 1.35e9T^{2}
71 1+2.95e4iT−1.80e9T2 1 + 2.95e4iT - 1.80e9T^{2}
73 1−5.43e4iT−2.07e9T2 1 - 5.43e4iT - 2.07e9T^{2}
79 1−8.69e4iT−3.07e9T2 1 - 8.69e4iT - 3.07e9T^{2}
83 1−5.07e4T+3.93e9T2 1 - 5.07e4T + 3.93e9T^{2}
89 1+2.16e4iT−5.58e9T2 1 + 2.16e4iT - 5.58e9T^{2}
97 1−4.60e3iT−8.58e9T2 1 - 4.60e3iT - 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

−11.10680642571416892728629592340, −10.60959335819753039035655333236, −10.29261188527787930677842400323, −8.602561825466709710017050985228, −7.32878637896082123247405620239, −6.46233979988778082498588790008, −5.68478279950016108007108939676, −3.86929578838428002571764211541, −2.07856697561917344190642678616, −0.43126290116962494598141457186, 0.53512234954362746254414966515, 1.75047927645228496508674710433, 4.49744371049738628762846180652, 5.48907396848420525809489886772, 6.46250955907779138377406378807, 7.34918221088904000528831225102, 8.854189234321673167395621106587, 9.568450919977591254188027637982, 10.83315790401838923868671559591, 11.43449405017763717233283341832

Graph of the ZZ-function along the critical line