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Properties

Label 2-14e2-196.111-c1-0-12
Degree 22
Conductor 196196
Sign 0.994+0.105i0.994 + 0.105i
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  + (−1.33 + 0.452i)2-s + (0.470 + 0.589i)3-s + (1.58 − 1.21i)4-s + (0.701 − 0.559i)5-s + (−0.897 − 0.577i)6-s + (0.938 − 2.47i)7-s + (−1.57 + 2.34i)8-s + (0.540 − 2.37i)9-s + (−0.686 + 1.06i)10-s + (0.406 − 0.0927i)11-s + (1.46 + 0.366i)12-s + (0.0704 − 0.0160i)13-s + (−0.137 + 3.73i)14-s + (0.660 + 0.150i)15-s + (1.05 − 3.85i)16-s + (−1.30 + 2.70i)17-s + ⋯
L(s)  = 1  + (−0.947 + 0.320i)2-s + (0.271 + 0.340i)3-s + (0.794 − 0.606i)4-s + (0.313 − 0.250i)5-s + (−0.366 − 0.235i)6-s + (0.354 − 0.934i)7-s + (−0.558 + 0.829i)8-s + (0.180 − 0.790i)9-s + (−0.217 + 0.337i)10-s + (0.122 − 0.0279i)11-s + (0.422 + 0.105i)12-s + (0.0195 − 0.00445i)13-s + (−0.0366 + 0.999i)14-s + (0.170 + 0.0389i)15-s + (0.263 − 0.964i)16-s + (−0.316 + 0.656i)17-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((0.994+0.105i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+1/2) L(s)=((0.994+0.105i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: 0.994+0.105i0.994 + 0.105i
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), 0.994+0.105i)(2,\ 196,\ (\ :1/2),\ 0.994 + 0.105i)

Particular Values

L(1)L(1) ≈\approx 0.947689−0.0499503i0.947689 - 0.0499503i
L(12)L(\frac12) ≈\approx 0.947689−0.0499503i0.947689 - 0.0499503i
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+(1.33−0.452i)T 1 + (1.33 - 0.452i)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

−12.29218153309336310389189110159, −11.22311448185571559134128861453, −10.24526121708785059591448516431, −9.501644299277822864245442122649, −8.604759742936891403882818410644, −7.47693572312119605774933910723, −6.53181380325021164799343397004, −5.13546435365586695112786578327, −3.48527108383320818407154633303, −1.32932671818058325214878930920, 1.84716201218990658252644485916, 2.98861605034068176941400770108, 5.15143408960262151496553532982, 6.60774425712436047843791495798, 7.67624092930279251237976326060, 8.525081729006280288544820939550, 9.519014285474332076490224673123, 10.42945584914771724908050034675, 11.56853693591519731354939829271, 12.17353013544937860850913899319

Graph of the ZZ-function along the critical line