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Properties

Label 2-14e2-196.111-c1-0-19
Degree 22
Conductor 196196
Sign 0.871+0.489i0.871 + 0.489i
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  + (1.40 − 0.173i)2-s + (−0.470 − 0.589i)3-s + (1.94 − 0.486i)4-s + (0.701 − 0.559i)5-s + (−0.762 − 0.746i)6-s + (−0.938 + 2.47i)7-s + (2.63 − 1.01i)8-s + (0.540 − 2.37i)9-s + (0.888 − 0.907i)10-s + (−0.406 + 0.0927i)11-s + (−1.19 − 0.915i)12-s + (0.0704 − 0.0160i)13-s + (−0.889 + 3.63i)14-s + (−0.660 − 0.150i)15-s + (3.52 − 1.88i)16-s + (−1.30 + 2.70i)17-s + ⋯
L(s)  = 1  + (0.992 − 0.122i)2-s + (−0.271 − 0.340i)3-s + (0.970 − 0.243i)4-s + (0.313 − 0.250i)5-s + (−0.311 − 0.304i)6-s + (−0.354 + 0.934i)7-s + (0.932 − 0.359i)8-s + (0.180 − 0.790i)9-s + (0.280 − 0.286i)10-s + (−0.122 + 0.0279i)11-s + (−0.346 − 0.264i)12-s + (0.0195 − 0.00445i)13-s + (−0.237 + 0.971i)14-s + (−0.170 − 0.0389i)15-s + (0.881 − 0.471i)16-s + (−0.316 + 0.656i)17-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((0.871+0.489i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+1/2) L(s)=((0.871+0.489i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: 0.871+0.489i0.871 + 0.489i
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), 0.871+0.489i)(2,\ 196,\ (\ :1/2),\ 0.871 + 0.489i)

Particular Values

L(1)L(1) ≈\approx 1.88417−0.493081i1.88417 - 0.493081i
L(12)L(\frac12) ≈\approx 1.88417−0.493081i1.88417 - 0.493081i
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+(−1.40+0.173i)T 1 + (-1.40 + 0.173i)T
7 1+(0.938−2.47i)T 1 + (0.938 - 2.47i)T
good3 1+(0.470+0.589i)T+(−0.667+2.92i)T2 1 + (0.470 + 0.589i)T + (-0.667 + 2.92i)T^{2}
5 1+(−0.701+0.559i)T+(1.11−4.87i)T2 1 + (-0.701 + 0.559i)T + (1.11 - 4.87i)T^{2}
11 1+(0.406−0.0927i)T+(9.91−4.77i)T2 1 + (0.406 - 0.0927i)T + (9.91 - 4.77i)T^{2}
13 1+(−0.0704+0.0160i)T+(11.7−5.64i)T2 1 + (-0.0704 + 0.0160i)T + (11.7 - 5.64i)T^{2}
17 1+(1.30−2.70i)T+(−10.5−13.2i)T2 1 + (1.30 - 2.70i)T + (-10.5 - 13.2i)T^{2}
19 1+7.07T+19T2 1 + 7.07T + 19T^{2}
23 1+(−0.339−0.706i)T+(−14.3+17.9i)T2 1 + (-0.339 - 0.706i)T + (-14.3 + 17.9i)T^{2}
29 1+(−4.62−2.22i)T+(18.0+22.6i)T2 1 + (-4.62 - 2.22i)T + (18.0 + 22.6i)T^{2}
31 1−2.54T+31T2 1 - 2.54T + 31T^{2}
37 1+(7.13+3.43i)T+(23.0+28.9i)T2 1 + (7.13 + 3.43i)T + (23.0 + 28.9i)T^{2}
41 1+(5.37−4.28i)T+(9.12−39.9i)T2 1 + (5.37 - 4.28i)T + (9.12 - 39.9i)T^{2}
43 1+(−3.75−2.99i)T+(9.56+41.9i)T2 1 + (-3.75 - 2.99i)T + (9.56 + 41.9i)T^{2}
47 1+(0.549+2.40i)T+(−42.3+20.3i)T2 1 + (0.549 + 2.40i)T + (-42.3 + 20.3i)T^{2}
53 1+(5.53−2.66i)T+(33.0−41.4i)T2 1 + (5.53 - 2.66i)T + (33.0 - 41.4i)T^{2}
59 1+(−6.25+7.84i)T+(−13.1−57.5i)T2 1 + (-6.25 + 7.84i)T + (-13.1 - 57.5i)T^{2}
61 1+(−5.42+11.2i)T+(−38.0−47.6i)T2 1 + (-5.42 + 11.2i)T + (-38.0 - 47.6i)T^{2}
67 1+10.3iT−67T2 1 + 10.3iT - 67T^{2}
71 1+(−3.56−7.40i)T+(−44.2+55.5i)T2 1 + (-3.56 - 7.40i)T + (-44.2 + 55.5i)T^{2}
73 1+(−11.0−2.52i)T+(65.7+31.6i)T2 1 + (-11.0 - 2.52i)T + (65.7 + 31.6i)T^{2}
79 1+8.93iT−79T2 1 + 8.93iT - 79T^{2}
83 1+(0.295−1.29i)T+(−74.7−36.0i)T2 1 + (0.295 - 1.29i)T + (-74.7 - 36.0i)T^{2}
89 1+(−3.97−0.906i)T+(80.1+38.6i)T2 1 + (-3.97 - 0.906i)T + (80.1 + 38.6i)T^{2}
97 1−12.4iT−97T2 1 - 12.4iT - 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

−12.69908712889988580060794344899, −11.77180789922614645923417219020, −10.73530279993332676426557844172, −9.554714235403499093949160292713, −8.371188730744883722003111973377, −6.72059300216607524674509267186, −6.14963027578595144096509393403, −5.00817065810136705602475835270, −3.54081104856381181197745571369, −1.96789192094837810030991889595, 2.41076258058828814050132025864, 4.03717959705686027450843392956, 4.92601323255985730878188095979, 6.28133916809896860464113566023, 7.10801075186352128246875888143, 8.339931088187970050713209752455, 10.23974695503839588955374573177, 10.52894942277785424312119475991, 11.64335349726221773217343307181, 12.79509701625209941065141745762

Graph of the ZZ-function along the critical line