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Properties

Label 2-14e2-196.111-c1-0-15
Degree 22
Conductor 196196
Sign 0.907−0.420i0.907 - 0.420i
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  + (1.38 − 0.287i)2-s + (1.31 + 1.64i)3-s + (1.83 − 0.795i)4-s + (−2.15 + 1.71i)5-s + (2.28 + 1.90i)6-s + (−0.0757 − 2.64i)7-s + (2.31 − 1.62i)8-s + (−0.317 + 1.38i)9-s + (−2.48 + 2.99i)10-s + (−5.04 + 1.15i)11-s + (3.71 + 1.97i)12-s + (4.41 − 1.00i)13-s + (−0.864 − 3.64i)14-s + (−5.64 − 1.28i)15-s + (2.73 − 2.91i)16-s + (−1.55 + 3.23i)17-s + ⋯
L(s)  = 1  + (0.979 − 0.203i)2-s + (0.757 + 0.949i)3-s + (0.917 − 0.397i)4-s + (−0.962 + 0.767i)5-s + (0.934 + 0.775i)6-s + (−0.0286 − 0.999i)7-s + (0.817 − 0.575i)8-s + (−0.105 + 0.463i)9-s + (−0.786 + 0.946i)10-s + (−1.52 + 0.347i)11-s + (1.07 + 0.570i)12-s + (1.22 − 0.279i)13-s + (−0.231 − 0.972i)14-s + (−1.45 − 0.332i)15-s + (0.683 − 0.729i)16-s + (−0.377 + 0.784i)17-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((0.907−0.420i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+1/2) L(s)=((0.907−0.420i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: 0.907−0.420i0.907 - 0.420i
Analytic conductor: 1.565061.56506
Root analytic conductor: 1.251021.25102
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ196(111,⋅)\chi_{196} (111, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 196, ( :1/2), 0.907−0.420i)(2,\ 196,\ (\ :1/2),\ 0.907 - 0.420i)

Particular Values

L(1)L(1) ≈\approx 2.12479+0.468665i2.12479 + 0.468665i
L(12)L(\frac12) ≈\approx 2.12479+0.468665i2.12479 + 0.468665i
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+(−1.38+0.287i)T 1 + (-1.38 + 0.287i)T
7 1+(0.0757+2.64i)T 1 + (0.0757 + 2.64i)T
good3 1+(−1.31−1.64i)T+(−0.667+2.92i)T2 1 + (-1.31 - 1.64i)T + (-0.667 + 2.92i)T^{2}
5 1+(2.15−1.71i)T+(1.11−4.87i)T2 1 + (2.15 - 1.71i)T + (1.11 - 4.87i)T^{2}
11 1+(5.04−1.15i)T+(9.91−4.77i)T2 1 + (5.04 - 1.15i)T + (9.91 - 4.77i)T^{2}
13 1+(−4.41+1.00i)T+(11.7−5.64i)T2 1 + (-4.41 + 1.00i)T + (11.7 - 5.64i)T^{2}
17 1+(1.55−3.23i)T+(−10.5−13.2i)T2 1 + (1.55 - 3.23i)T + (-10.5 - 13.2i)T^{2}
19 1+1.68T+19T2 1 + 1.68T + 19T^{2}
23 1+(3.00+6.24i)T+(−14.3+17.9i)T2 1 + (3.00 + 6.24i)T + (-14.3 + 17.9i)T^{2}
29 1+(2.62+1.26i)T+(18.0+22.6i)T2 1 + (2.62 + 1.26i)T + (18.0 + 22.6i)T^{2}
31 1+2.18T+31T2 1 + 2.18T + 31T^{2}
37 1+(−9.60−4.62i)T+(23.0+28.9i)T2 1 + (-9.60 - 4.62i)T + (23.0 + 28.9i)T^{2}
41 1+(2.45−1.95i)T+(9.12−39.9i)T2 1 + (2.45 - 1.95i)T + (9.12 - 39.9i)T^{2}
43 1+(0.804+0.641i)T+(9.56+41.9i)T2 1 + (0.804 + 0.641i)T + (9.56 + 41.9i)T^{2}
47 1+(−0.583−2.55i)T+(−42.3+20.3i)T2 1 + (-0.583 - 2.55i)T + (-42.3 + 20.3i)T^{2}
53 1+(−3.85+1.85i)T+(33.0−41.4i)T2 1 + (-3.85 + 1.85i)T + (33.0 - 41.4i)T^{2}
59 1+(1.51−1.89i)T+(−13.1−57.5i)T2 1 + (1.51 - 1.89i)T + (-13.1 - 57.5i)T^{2}
61 1+(5.08−10.5i)T+(−38.0−47.6i)T2 1 + (5.08 - 10.5i)T + (-38.0 - 47.6i)T^{2}
67 1−13.2iT−67T2 1 - 13.2iT - 67T^{2}
71 1+(−1.49−3.11i)T+(−44.2+55.5i)T2 1 + (-1.49 - 3.11i)T + (-44.2 + 55.5i)T^{2}
73 1+(8.97+2.04i)T+(65.7+31.6i)T2 1 + (8.97 + 2.04i)T + (65.7 + 31.6i)T^{2}
79 1+4.00iT−79T2 1 + 4.00iT - 79T^{2}
83 1+(−0.261+1.14i)T+(−74.7−36.0i)T2 1 + (-0.261 + 1.14i)T + (-74.7 - 36.0i)T^{2}
89 1+(−12.0−2.76i)T+(80.1+38.6i)T2 1 + (-12.0 - 2.76i)T + (80.1 + 38.6i)T^{2}
97 1−2.97iT−97T2 1 - 2.97iT - 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

−12.86185839250867776243073515528, −11.38420528862572209574488290706, −10.52934852933545762662806913607, −10.23783946503593984143939958364, −8.345878045951090455916595388838, −7.48477219489677528835765266982, −6.22452993246934717594738415941, −4.45878624473715993772287825580, −3.83683576308199453596817950109, −2.81234615364351205431316384111, 2.14671551897986856252501679973, 3.43799245975935930841323189450, 4.94788556245965909363779751676, 6.04942203323068915735135049436, 7.54818403133363915328290555878, 8.071672591050218683219055196361, 8.942266202205957047475819263724, 10.99703248067377532726731016131, 11.79834764411887281983610897999, 12.76915601800572486065884861173

Graph of the ZZ-function along the critical line