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Properties

Label 2-14e2-196.111-c1-0-1
Degree 22
Conductor 196196
Sign 0.219−0.975i0.219 - 0.975i
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  + (−0.368 − 1.36i)2-s + (−1.56 − 1.95i)3-s + (−1.72 + 1.00i)4-s + (−2.80 + 2.23i)5-s + (−2.09 + 2.85i)6-s + (2.40 − 1.09i)7-s + (2.00 + 1.99i)8-s + (−0.728 + 3.19i)9-s + (4.08 + 3.00i)10-s + (−3.69 + 0.843i)11-s + (4.66 + 1.81i)12-s + (−0.397 + 0.0908i)13-s + (−2.38 − 2.88i)14-s + (8.76 + 2.00i)15-s + (1.97 − 3.47i)16-s + (−1.99 + 4.14i)17-s + ⋯
L(s)  = 1  + (−0.260 − 0.965i)2-s + (−0.901 − 1.13i)3-s + (−0.864 + 0.502i)4-s + (−1.25 + 1.00i)5-s + (−0.856 + 1.16i)6-s + (0.909 − 0.414i)7-s + (0.710 + 0.703i)8-s + (−0.242 + 1.06i)9-s + (1.29 + 0.951i)10-s + (−1.11 + 0.254i)11-s + (1.34 + 0.523i)12-s + (−0.110 + 0.0251i)13-s + (−0.637 − 0.770i)14-s + (2.26 + 0.516i)15-s + (0.494 − 0.869i)16-s + (−0.483 + 1.00i)17-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: 0.219−0.975i0.219 - 0.975i
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), 0.219−0.975i)(2,\ 196,\ (\ :1/2),\ 0.219 - 0.975i)

Particular Values

L(1)L(1) ≈\approx 0.0545740+0.0436740i0.0545740 + 0.0436740i
L(12)L(\frac12) ≈\approx 0.0545740+0.0436740i0.0545740 + 0.0436740i
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+(0.368+1.36i)T 1 + (0.368 + 1.36i)T
7 1+(−2.40+1.09i)T 1 + (-2.40 + 1.09i)T
good3 1+(1.56+1.95i)T+(−0.667+2.92i)T2 1 + (1.56 + 1.95i)T + (-0.667 + 2.92i)T^{2}
5 1+(2.80−2.23i)T+(1.11−4.87i)T2 1 + (2.80 - 2.23i)T + (1.11 - 4.87i)T^{2}
11 1+(3.69−0.843i)T+(9.91−4.77i)T2 1 + (3.69 - 0.843i)T + (9.91 - 4.77i)T^{2}
13 1+(0.397−0.0908i)T+(11.7−5.64i)T2 1 + (0.397 - 0.0908i)T + (11.7 - 5.64i)T^{2}
17 1+(1.99−4.14i)T+(−10.5−13.2i)T2 1 + (1.99 - 4.14i)T + (-10.5 - 13.2i)T^{2}
19 1+3.39T+19T2 1 + 3.39T + 19T^{2}
23 1+(−0.764−1.58i)T+(−14.3+17.9i)T2 1 + (-0.764 - 1.58i)T + (-14.3 + 17.9i)T^{2}
29 1+(−3.44−1.65i)T+(18.0+22.6i)T2 1 + (-3.44 - 1.65i)T + (18.0 + 22.6i)T^{2}
31 1+10.4T+31T2 1 + 10.4T + 31T^{2}
37 1+(4.97+2.39i)T+(23.0+28.9i)T2 1 + (4.97 + 2.39i)T + (23.0 + 28.9i)T^{2}
41 1+(6.23−4.97i)T+(9.12−39.9i)T2 1 + (6.23 - 4.97i)T + (9.12 - 39.9i)T^{2}
43 1+(−1.08−0.861i)T+(9.56+41.9i)T2 1 + (-1.08 - 0.861i)T + (9.56 + 41.9i)T^{2}
47 1+(1.15+5.04i)T+(−42.3+20.3i)T2 1 + (1.15 + 5.04i)T + (-42.3 + 20.3i)T^{2}
53 1+(1.01−0.489i)T+(33.0−41.4i)T2 1 + (1.01 - 0.489i)T + (33.0 - 41.4i)T^{2}
59 1+(−6.97+8.74i)T+(−13.1−57.5i)T2 1 + (-6.97 + 8.74i)T + (-13.1 - 57.5i)T^{2}
61 1+(−3.88+8.07i)T+(−38.0−47.6i)T2 1 + (-3.88 + 8.07i)T + (-38.0 - 47.6i)T^{2}
67 1−5.92iT−67T2 1 - 5.92iT - 67T^{2}
71 1+(6.36+13.2i)T+(−44.2+55.5i)T2 1 + (6.36 + 13.2i)T + (-44.2 + 55.5i)T^{2}
73 1+(−2.53−0.577i)T+(65.7+31.6i)T2 1 + (-2.53 - 0.577i)T + (65.7 + 31.6i)T^{2}
79 1−5.63iT−79T2 1 - 5.63iT - 79T^{2}
83 1+(1.68−7.39i)T+(−74.7−36.0i)T2 1 + (1.68 - 7.39i)T + (-74.7 - 36.0i)T^{2}
89 1+(1.80+0.412i)T+(80.1+38.6i)T2 1 + (1.80 + 0.412i)T + (80.1 + 38.6i)T^{2}
97 1+1.47iT−97T2 1 + 1.47iT - 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

−12.50374430923350875200872568738, −11.56101485183802465325166326425, −10.96937456442211891043464037255, −10.45176754514252179396541752792, −8.374959938551320844948381440305, −7.66694895980453917500118517843, −6.86174424459537540153358739944, −5.10667739076210688426671499310, −3.72775860130623268302482811147, −1.97508749739974956318094550168, 0.07356055962894261887860081863, 4.18026481827591657747882959626, 4.93646849580258050938892439579, 5.51414785263389617328117657678, 7.31292284230162527226556120079, 8.359531043849727880438867577786, 9.003015287452066897193022226692, 10.37744765769235518016264614320, 11.18338301481807514329531773808, 12.11925523179493761012302212454

Graph of the ZZ-function along the critical line