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Properties

Label 2-14e2-7.4-c3-0-9
Degree 22
Conductor 196196
Sign −0.991+0.126i-0.991 + 0.126i
Analytic cond. 11.564311.5643
Root an. cond. 3.400643.40064
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.04 − 5.26i)3-s + (−9.58 − 16.5i)5-s + (−5 − 8.66i)9-s + (13.2 − 23.0i)11-s − 10.3·13-s − 116.·15-s + (−50.6 + 87.7i)17-s + (−46.8 − 81.1i)19-s + (4.28 + 7.42i)23-s + (−121. + 209. i)25-s + 103.·27-s + 52.3·29-s + (−27.7 + 47.9i)31-s + (−80.8 − 140. i)33-s + (−214. − 371. i)37-s + ⋯
L(s)  = 1  + (0.585 − 1.01i)3-s + (−0.857 − 1.48i)5-s + (−0.185 − 0.320i)9-s + (0.364 − 0.630i)11-s − 0.220·13-s − 2.00·15-s + (−0.722 + 1.25i)17-s + (−0.566 − 0.980i)19-s + (0.0388 + 0.0673i)23-s + (−0.969 + 1.67i)25-s + 0.737·27-s + 0.335·29-s + (−0.160 + 0.278i)31-s + (−0.426 − 0.738i)33-s + (−0.951 − 1.64i)37-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((−0.991+0.126i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+3/2) L(s)=((−0.991+0.126i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: −0.991+0.126i-0.991 + 0.126i
Analytic conductor: 11.564311.5643
Root analytic conductor: 3.400643.40064
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ196(165,⋅)\chi_{196} (165, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 196, ( :3/2), −0.991+0.126i)(2,\ 196,\ (\ :3/2),\ -0.991 + 0.126i)

Particular Values

L(2)L(2) ≈\approx 0.0841014−1.32526i0.0841014 - 1.32526i
L(12)L(\frac12) ≈\approx 0.0841014−1.32526i0.0841014 - 1.32526i
L(52)L(\frac{5}{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
7 1 1
good3 1+(−3.04+5.26i)T+(−13.5−23.3i)T2 1 + (-3.04 + 5.26i)T + (-13.5 - 23.3i)T^{2}
5 1+(9.58+16.5i)T+(−62.5+108.i)T2 1 + (9.58 + 16.5i)T + (-62.5 + 108. i)T^{2}
11 1+(−13.2+23.0i)T+(−665.5−1.15e3i)T2 1 + (-13.2 + 23.0i)T + (-665.5 - 1.15e3i)T^{2}
13 1+10.3T+2.19e3T2 1 + 10.3T + 2.19e3T^{2}
17 1+(50.6−87.7i)T+(−2.45e3−4.25e3i)T2 1 + (50.6 - 87.7i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(46.8+81.1i)T+(−3.42e3+5.94e3i)T2 1 + (46.8 + 81.1i)T + (-3.42e3 + 5.94e3i)T^{2}
23 1+(−4.28−7.42i)T+(−6.08e3+1.05e4i)T2 1 + (-4.28 - 7.42i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1−52.3T+2.43e4T2 1 - 52.3T + 2.43e4T^{2}
31 1+(27.7−47.9i)T+(−1.48e4−2.57e4i)T2 1 + (27.7 - 47.9i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+(214.+371.i)T+(−2.53e4+4.38e4i)T2 1 + (214. + 371. i)T + (-2.53e4 + 4.38e4i)T^{2}
41 1−137.T+6.89e4T2 1 - 137.T + 6.89e4T^{2}
43 1+172T+7.95e4T2 1 + 172T + 7.95e4T^{2}
47 1+(24.6+42.6i)T+(−5.19e4+8.99e4i)T2 1 + (24.6 + 42.6i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1+(−237.+410.i)T+(−7.44e4−1.28e5i)T2 1 + (-237. + 410. i)T + (-7.44e4 - 1.28e5i)T^{2}
59 1+(−98.5+170.i)T+(−1.02e5−1.77e5i)T2 1 + (-98.5 + 170. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(200.+347.i)T+(−1.13e5+1.96e5i)T2 1 + (200. + 347. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(62.7−108.i)T+(−1.50e5−2.60e5i)T2 1 + (62.7 - 108. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1−788.T+3.57e5T2 1 - 788.T + 3.57e5T^{2}
73 1+(−302.+523.i)T+(−1.94e5−3.36e5i)T2 1 + (-302. + 523. i)T + (-1.94e5 - 3.36e5i)T^{2}
79 1+(391.+678.i)T+(−2.46e5+4.26e5i)T2 1 + (391. + 678. i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1−339.T+5.71e5T2 1 - 339.T + 5.71e5T^{2}
89 1+(−255.−443.i)T+(−3.52e5+6.10e5i)T2 1 + (-255. - 443. i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1−672.T+9.12e5T2 1 - 672.T + 9.12e5T^{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

−11.88029847280773071799838801912, −10.79138802123928277855956213700, −8.998483877564591449168001447467, −8.554584837587603875343612247253, −7.70200789183751556920023823076, −6.54804759597979250091830446839, −4.99180013383635942101988857785, −3.79842680354324784721103833044, −1.91581407004642967892325115095, −0.53431010427785656541803549222, 2.64256447565118468292692028749, 3.67666670266718767960509032568, 4.61284394106216474085501005039, 6.54333274822389427129270429305, 7.37765048366936998595917940608, 8.582114420886776112576849923702, 9.762733666493983781744629246069, 10.41017752248029105218107090781, 11.39189742909800507407235907304, 12.24770254019677880506097192400

Graph of the ZZ-function along the critical line