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Properties

Label 2-14e2-28.27-c5-0-22
Degree 22
Conductor 196196
Sign −0.396−0.917i-0.396 - 0.917i
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  + (1.57 + 5.43i)2-s − 7.64·3-s + (−27.0 + 17.1i)4-s − 63.9i·5-s + (−12.0 − 41.5i)6-s + (−135. − 119. i)8-s − 184.·9-s + (347. − 100. i)10-s − 601. i·11-s + (206. − 130. i)12-s + 890. i·13-s + 488. i·15-s + (436. − 926. i)16-s + 116. i·17-s + (−291. − 1.00e3i)18-s + 901.·19-s + ⋯
L(s)  = 1  + (0.278 + 0.960i)2-s − 0.490·3-s + (−0.844 + 0.535i)4-s − 1.14i·5-s + (−0.136 − 0.470i)6-s + (−0.749 − 0.661i)8-s − 0.759·9-s + (1.09 − 0.318i)10-s − 1.49i·11-s + (0.414 − 0.262i)12-s + 1.46i·13-s + 0.560i·15-s + (0.426 − 0.904i)16-s + 0.0978i·17-s + (−0.211 − 0.729i)18-s + 0.573·19-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((−0.396−0.917i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.917i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+5/2) L(s)=((−0.396−0.917i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.396 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: −0.396−0.917i-0.396 - 0.917i
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), −0.396−0.917i)(2,\ 196,\ (\ :5/2),\ -0.396 - 0.917i)

Particular Values

L(3)L(3) ≈\approx 1.0859951771.085995177
L(12)L(\frac12) ≈\approx 1.0859951771.085995177
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+(−1.57−5.43i)T 1 + (-1.57 - 5.43i)T
7 1 1
good3 1+7.64T+243T2 1 + 7.64T + 243T^{2}
5 1+63.9iT−3.12e3T2 1 + 63.9iT - 3.12e3T^{2}
11 1+601.iT−1.61e5T2 1 + 601. iT - 1.61e5T^{2}
13 1−890.iT−3.71e5T2 1 - 890. iT - 3.71e5T^{2}
17 1−116.iT−1.41e6T2 1 - 116. iT - 1.41e6T^{2}
19 1−901.T+2.47e6T2 1 - 901.T + 2.47e6T^{2}
23 1−3.88e3iT−6.43e6T2 1 - 3.88e3iT - 6.43e6T^{2}
29 1−124.T+2.05e7T2 1 - 124.T + 2.05e7T^{2}
31 1+7.11T+2.86e7T2 1 + 7.11T + 2.86e7T^{2}
37 1−1.50e3T+6.93e7T2 1 - 1.50e3T + 6.93e7T^{2}
41 1−1.93e4iT−1.15e8T2 1 - 1.93e4iT - 1.15e8T^{2}
43 1+2.30e3iT−1.47e8T2 1 + 2.30e3iT - 1.47e8T^{2}
47 1+8.62e3T+2.29e8T2 1 + 8.62e3T + 2.29e8T^{2}
53 1+2.81e4T+4.18e8T2 1 + 2.81e4T + 4.18e8T^{2}
59 1−5.26e4T+7.14e8T2 1 - 5.26e4T + 7.14e8T^{2}
61 1−2.39e4iT−8.44e8T2 1 - 2.39e4iT - 8.44e8T^{2}
67 1+4.95e4iT−1.35e9T2 1 + 4.95e4iT - 1.35e9T^{2}
71 1−6.72e4iT−1.80e9T2 1 - 6.72e4iT - 1.80e9T^{2}
73 1+3.96e4iT−2.07e9T2 1 + 3.96e4iT - 2.07e9T^{2}
79 1+5.60e4iT−3.07e9T2 1 + 5.60e4iT - 3.07e9T^{2}
83 1−1.42e4T+3.93e9T2 1 - 1.42e4T + 3.93e9T^{2}
89 1−1.19e5iT−5.58e9T2 1 - 1.19e5iT - 5.58e9T^{2}
97 1−5.12e4iT−8.58e9T2 1 - 5.12e4iT - 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

−11.87233925823795715813503386961, −11.36252788807899249368540324517, −9.479556658490599227401086332961, −8.791404845899280847693358270828, −7.954128268491671785299760844120, −6.49102384024320324715806839407, −5.62619729346174372133916884929, −4.78748806528215245130252010779, −3.43739564416935754928894537421, −0.958844049510372326062902434849, 0.41987500096200220108342475654, 2.32588440946119157417094683994, 3.24191351276746786130093548786, 4.77389698495591046463947933804, 5.82505388786763187386612448502, 7.04189954314529672885548203758, 8.403049599989077376035090595444, 9.846200398387965027757524390729, 10.49026189725845663808042123998, 11.21065235180953759858625225690

Graph of the ZZ-function along the critical line