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Properties

Label 2-14e2-28.27-c5-0-12
Degree 22
Conductor 196196
Sign −0.975+0.219i-0.975 + 0.219i
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  + (−0.565 + 5.62i)2-s + 16.9·3-s + (−31.3 − 6.36i)4-s − 9.31i·5-s + (−9.56 + 95.2i)6-s + (53.5 − 172. i)8-s + 43.5·9-s + (52.4 + 5.26i)10-s + 42.9i·11-s + (−530. − 107. i)12-s + 607. i·13-s − 157. i·15-s + (943. + 398. i)16-s + 568. i·17-s + (−24.5 + 244. i)18-s − 2.51e3·19-s + ⋯
L(s)  = 1  + (−0.0998 + 0.994i)2-s + 1.08·3-s + (−0.980 − 0.198i)4-s − 0.166i·5-s + (−0.108 + 1.08i)6-s + (0.295 − 0.955i)8-s + 0.179·9-s + (0.165 + 0.0166i)10-s + 0.107i·11-s + (−1.06 − 0.215i)12-s + 0.997i·13-s − 0.180i·15-s + (0.920 + 0.389i)16-s + 0.476i·17-s + (−0.0178 + 0.178i)18-s − 1.59·19-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((−0.975+0.219i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+5/2) L(s)=((−0.975+0.219i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: −0.975+0.219i-0.975 + 0.219i
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), −0.975+0.219i)(2,\ 196,\ (\ :5/2),\ -0.975 + 0.219i)

Particular Values

L(3)L(3) ≈\approx 1.1070318901.107031890
L(12)L(\frac12) ≈\approx 1.1070318901.107031890
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+(0.565−5.62i)T 1 + (0.565 - 5.62i)T
7 1 1
good3 1−16.9T+243T2 1 - 16.9T + 243T^{2}
5 1+9.31iT−3.12e3T2 1 + 9.31iT - 3.12e3T^{2}
11 1−42.9iT−1.61e5T2 1 - 42.9iT - 1.61e5T^{2}
13 1−607.iT−3.71e5T2 1 - 607. iT - 3.71e5T^{2}
17 1−568.iT−1.41e6T2 1 - 568. iT - 1.41e6T^{2}
19 1+2.51e3T+2.47e6T2 1 + 2.51e3T + 2.47e6T^{2}
23 1−2.05e3iT−6.43e6T2 1 - 2.05e3iT - 6.43e6T^{2}
29 1+6.87e3T+2.05e7T2 1 + 6.87e3T + 2.05e7T^{2}
31 1−2.40e3T+2.86e7T2 1 - 2.40e3T + 2.86e7T^{2}
37 1+1.49e4T+6.93e7T2 1 + 1.49e4T + 6.93e7T^{2}
41 1−1.41e4iT−1.15e8T2 1 - 1.41e4iT - 1.15e8T^{2}
43 1−1.26e4iT−1.47e8T2 1 - 1.26e4iT - 1.47e8T^{2}
47 1−1.25e4T+2.29e8T2 1 - 1.25e4T + 2.29e8T^{2}
53 1−1.02e4T+4.18e8T2 1 - 1.02e4T + 4.18e8T^{2}
59 1+2.47e4T+7.14e8T2 1 + 2.47e4T + 7.14e8T^{2}
61 1+3.97e4iT−8.44e8T2 1 + 3.97e4iT - 8.44e8T^{2}
67 1+2.19e4iT−1.35e9T2 1 + 2.19e4iT - 1.35e9T^{2}
71 1+1.04e4iT−1.80e9T2 1 + 1.04e4iT - 1.80e9T^{2}
73 1+3.84e4iT−2.07e9T2 1 + 3.84e4iT - 2.07e9T^{2}
79 1−8.28e4iT−3.07e9T2 1 - 8.28e4iT - 3.07e9T^{2}
83 1+5.23e4T+3.93e9T2 1 + 5.23e4T + 3.93e9T^{2}
89 1−384.iT−5.58e9T2 1 - 384. iT - 5.58e9T^{2}
97 1+4.95e4iT−8.58e9T2 1 + 4.95e4iT - 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

−12.53859855652110624632350258291, −10.99413798474761649856444018109, −9.660940785598896926109778813915, −8.892379337504356349862242181062, −8.231723579394378388248042347194, −7.15642659103070537404398454875, −6.08515433526271049856211668720, −4.64702201418334333865759383642, −3.54926655874802376504461004823, −1.79945909854449610517676098815, 0.28861165551966822362461551757, 2.07684002568769426750330969968, 2.99744075848361015432027199012, 4.05188376357100386401539675475, 5.51202424749800468776633527411, 7.31321306132347918169724185050, 8.560108494408197365317662176701, 8.892459543191374243489399494112, 10.26281641437967542518795866873, 10.85520847318506779723923110082

Graph of the ZZ-function along the critical line