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Properties

Label 2-14e2-196.103-c1-0-17
Degree 22
Conductor 196196
Sign 0.892+0.450i0.892 + 0.450i
Analytic cond. 1.565061.56506
Root an. cond. 1.251021.25102
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  + (0.301 − 1.38i)2-s + (2.71 + 1.85i)3-s + (−1.81 − 0.832i)4-s + (1.96 + 0.147i)5-s + (3.37 − 3.19i)6-s + (−2.22 − 1.42i)7-s + (−1.69 + 2.26i)8-s + (2.85 + 7.26i)9-s + (0.794 − 2.66i)10-s + (−4.60 − 1.80i)11-s + (−3.39 − 5.62i)12-s + (0.963 − 0.768i)13-s + (−2.64 + 2.64i)14-s + (5.06 + 4.03i)15-s + (2.61 + 3.02i)16-s + (−0.724 − 0.780i)17-s + ⋯
L(s)  = 1  + (0.212 − 0.977i)2-s + (1.56 + 1.06i)3-s + (−0.909 − 0.416i)4-s + (0.878 + 0.0658i)5-s + (1.37 − 1.30i)6-s + (−0.842 − 0.539i)7-s + (−0.600 + 0.799i)8-s + (0.950 + 2.42i)9-s + (0.251 − 0.844i)10-s + (−1.38 − 0.544i)11-s + (−0.980 − 1.62i)12-s + (0.267 − 0.213i)13-s + (−0.706 + 0.708i)14-s + (1.30 + 1.04i)15-s + (0.653 + 0.756i)16-s + (−0.175 − 0.189i)17-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((0.892+0.450i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.450i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+1/2) L(s)=((0.892+0.450i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 + 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: 0.892+0.450i0.892 + 0.450i
Analytic conductor: 1.565061.56506
Root analytic conductor: 1.251021.25102
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ196(103,⋅)\chi_{196} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 196, ( :1/2), 0.892+0.450i)(2,\ 196,\ (\ :1/2),\ 0.892 + 0.450i)

Particular Values

L(1)L(1) ≈\approx 1.82783−0.435318i1.82783 - 0.435318i
L(12)L(\frac12) ≈\approx 1.82783−0.435318i1.82783 - 0.435318i
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+(−0.301+1.38i)T 1 + (-0.301 + 1.38i)T
7 1+(2.22+1.42i)T 1 + (2.22 + 1.42i)T
good3 1+(−2.71−1.85i)T+(1.09+2.79i)T2 1 + (-2.71 - 1.85i)T + (1.09 + 2.79i)T^{2}
5 1+(−1.96−0.147i)T+(4.94+0.745i)T2 1 + (-1.96 - 0.147i)T + (4.94 + 0.745i)T^{2}
11 1+(4.60+1.80i)T+(8.06+7.48i)T2 1 + (4.60 + 1.80i)T + (8.06 + 7.48i)T^{2}
13 1+(−0.963+0.768i)T+(2.89−12.6i)T2 1 + (-0.963 + 0.768i)T + (2.89 - 12.6i)T^{2}
17 1+(0.724+0.780i)T+(−1.27+16.9i)T2 1 + (0.724 + 0.780i)T + (-1.27 + 16.9i)T^{2}
19 1+(−0.263+0.456i)T+(−9.5−16.4i)T2 1 + (-0.263 + 0.456i)T + (-9.5 - 16.4i)T^{2}
23 1+(−1.53+1.65i)T+(−1.71−22.9i)T2 1 + (-1.53 + 1.65i)T + (-1.71 - 22.9i)T^{2}
29 1+(0.514+2.25i)T+(−26.1+12.5i)T2 1 + (0.514 + 2.25i)T + (-26.1 + 12.5i)T^{2}
31 1+(2.23+3.87i)T+(−15.5+26.8i)T2 1 + (2.23 + 3.87i)T + (-15.5 + 26.8i)T^{2}
37 1+(−0.148−0.0457i)T+(30.5+20.8i)T2 1 + (-0.148 - 0.0457i)T + (30.5 + 20.8i)T^{2}
41 1+(−1.00−2.09i)T+(−25.5+32.0i)T2 1 + (-1.00 - 2.09i)T + (-25.5 + 32.0i)T^{2}
43 1+(4.03−8.37i)T+(−26.8−33.6i)T2 1 + (4.03 - 8.37i)T + (-26.8 - 33.6i)T^{2}
47 1+(−9.44+1.42i)T+(44.9−13.8i)T2 1 + (-9.44 + 1.42i)T + (44.9 - 13.8i)T^{2}
53 1+(2.10−0.649i)T+(43.7−29.8i)T2 1 + (2.10 - 0.649i)T + (43.7 - 29.8i)T^{2}
59 1+(−0.949−12.6i)T+(−58.3+8.79i)T2 1 + (-0.949 - 12.6i)T + (-58.3 + 8.79i)T^{2}
61 1+(0.310−1.00i)T+(−50.4−34.3i)T2 1 + (0.310 - 1.00i)T + (-50.4 - 34.3i)T^{2}
67 1+(0.898−0.518i)T+(33.5−58.0i)T2 1 + (0.898 - 0.518i)T + (33.5 - 58.0i)T^{2}
71 1+(−11.2−2.56i)T+(63.9+30.8i)T2 1 + (-11.2 - 2.56i)T + (63.9 + 30.8i)T^{2}
73 1+(−0.954+6.33i)T+(−69.7−21.5i)T2 1 + (-0.954 + 6.33i)T + (-69.7 - 21.5i)T^{2}
79 1+(7.88+4.55i)T+(39.5+68.4i)T2 1 + (7.88 + 4.55i)T + (39.5 + 68.4i)T^{2}
83 1+(−5.49+6.89i)T+(−18.4−80.9i)T2 1 + (-5.49 + 6.89i)T + (-18.4 - 80.9i)T^{2}
89 1+(8.70−3.41i)T+(65.2−60.5i)T2 1 + (8.70 - 3.41i)T + (65.2 - 60.5i)T^{2}
97 1+7.11iT−97T2 1 + 7.11iT - 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

−12.96822916284343413706175962986, −10.95859461731912923303279794887, −10.25442037080331314627091650995, −9.692664511005179806175362468255, −8.861091057941341384535290462141, −7.80004486433542395916095709296, −5.69449438497478913233933814356, −4.40426382767895965407178419320, −3.21178616441235297731123680062, −2.43537699583699612841144249481, 2.25169043259422713892605854663, 3.48077222779893102072130200406, 5.47157298235666362998483699341, 6.61132768257009392082003087886, 7.42973595922272093086660545191, 8.456834435733624741333393222930, 9.219351686913492345537844382541, 9.995152475199210160865063879600, 12.47611335969163568509339897080, 12.87139432608284534923228496976

Graph of the ZZ-function along the critical line