↑

Properties

Label 2-14e2-196.111-c1-0-13
Degree 22
Conductor 196196
Sign 0.999+0.000891i0.999 + 0.000891i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.13 − 0.846i)2-s + (1.79 + 2.25i)3-s + (0.566 + 1.91i)4-s + (2.89 − 2.30i)5-s + (−0.127 − 4.07i)6-s + (−1.61 − 2.09i)7-s + (0.982 − 2.65i)8-s + (−1.18 + 5.17i)9-s + (−5.22 + 0.164i)10-s + (1.04 − 0.238i)11-s + (−3.30 + 4.72i)12-s + (1.70 − 0.388i)13-s + (0.0583 + 3.74i)14-s + (10.3 + 2.37i)15-s + (−3.35 + 2.17i)16-s + (−2.15 + 4.47i)17-s + ⋯
L(s)  = 1  + (−0.801 − 0.598i)2-s + (1.03 + 1.30i)3-s + (0.283 + 0.959i)4-s + (1.29 − 1.03i)5-s + (−0.0522 − 1.66i)6-s + (−0.611 − 0.791i)7-s + (0.347 − 0.937i)8-s + (−0.393 + 1.72i)9-s + (−1.65 + 0.0519i)10-s + (0.314 − 0.0718i)11-s + (−0.953 + 1.36i)12-s + (0.472 − 0.107i)13-s + (0.0155 + 0.999i)14-s + (2.68 + 0.612i)15-s + (−0.839 + 0.543i)16-s + (−0.522 + 1.08i)17-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((0.999+0.000891i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.000891i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+1/2) L(s)=((0.999+0.000891i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.000891i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: 0.999+0.000891i0.999 + 0.000891i
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), 0.999+0.000891i)(2,\ 196,\ (\ :1/2),\ 0.999 + 0.000891i)

Particular Values

L(1)L(1) ≈\approx 1.26505−0.000563984i1.26505 - 0.000563984i
L(12)L(\frac12) ≈\approx 1.26505−0.000563984i1.26505 - 0.000563984i
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+(1.13+0.846i)T 1 + (1.13 + 0.846i)T
7 1+(1.61+2.09i)T 1 + (1.61 + 2.09i)T
good3 1+(−1.79−2.25i)T+(−0.667+2.92i)T2 1 + (-1.79 - 2.25i)T + (-0.667 + 2.92i)T^{2}
5 1+(−2.89+2.30i)T+(1.11−4.87i)T2 1 + (-2.89 + 2.30i)T + (1.11 - 4.87i)T^{2}
11 1+(−1.04+0.238i)T+(9.91−4.77i)T2 1 + (-1.04 + 0.238i)T + (9.91 - 4.77i)T^{2}
13 1+(−1.70+0.388i)T+(11.7−5.64i)T2 1 + (-1.70 + 0.388i)T + (11.7 - 5.64i)T^{2}
17 1+(2.15−4.47i)T+(−10.5−13.2i)T2 1 + (2.15 - 4.47i)T + (-10.5 - 13.2i)T^{2}
19 1+5.54T+19T2 1 + 5.54T + 19T^{2}
23 1+(−1.41−2.93i)T+(−14.3+17.9i)T2 1 + (-1.41 - 2.93i)T + (-14.3 + 17.9i)T^{2}
29 1+(3.37+1.62i)T+(18.0+22.6i)T2 1 + (3.37 + 1.62i)T + (18.0 + 22.6i)T^{2}
31 1+3.97T+31T2 1 + 3.97T + 31T^{2}
37 1+(−0.739−0.356i)T+(23.0+28.9i)T2 1 + (-0.739 - 0.356i)T + (23.0 + 28.9i)T^{2}
41 1+(−0.928+0.740i)T+(9.12−39.9i)T2 1 + (-0.928 + 0.740i)T + (9.12 - 39.9i)T^{2}
43 1+(0.119+0.0951i)T+(9.56+41.9i)T2 1 + (0.119 + 0.0951i)T + (9.56 + 41.9i)T^{2}
47 1+(−1.04−4.58i)T+(−42.3+20.3i)T2 1 + (-1.04 - 4.58i)T + (-42.3 + 20.3i)T^{2}
53 1+(9.27−4.46i)T+(33.0−41.4i)T2 1 + (9.27 - 4.46i)T + (33.0 - 41.4i)T^{2}
59 1+(−4.35+5.45i)T+(−13.1−57.5i)T2 1 + (-4.35 + 5.45i)T + (-13.1 - 57.5i)T^{2}
61 1+(−5.04+10.4i)T+(−38.0−47.6i)T2 1 + (-5.04 + 10.4i)T + (-38.0 - 47.6i)T^{2}
67 1+6.15iT−67T2 1 + 6.15iT - 67T^{2}
71 1+(2.60+5.40i)T+(−44.2+55.5i)T2 1 + (2.60 + 5.40i)T + (-44.2 + 55.5i)T^{2}
73 1+(12.6+2.88i)T+(65.7+31.6i)T2 1 + (12.6 + 2.88i)T + (65.7 + 31.6i)T^{2}
79 1+10.7iT−79T2 1 + 10.7iT - 79T^{2}
83 1+(2.65−11.6i)T+(−74.7−36.0i)T2 1 + (2.65 - 11.6i)T + (-74.7 - 36.0i)T^{2}
89 1+(−14.5−3.31i)T+(80.1+38.6i)T2 1 + (-14.5 - 3.31i)T + (80.1 + 38.6i)T^{2}
97 1−4.59iT−97T2 1 - 4.59iT - 97T^{2}
show more
show less
   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

−12.84425157384000821077339349703, −10.90827079367995036458550659264, −10.29898613527894119434942439536, −9.375713334429760347249992690121, −9.030317475108074522962995252790, −8.051794342334286903266478232720, −6.25559048209597197431060345043, −4.47455198529290718746763205287, −3.52847065345882393897485417549, −1.89772162623190874095172712851, 1.94652721179471041450947332512, 2.73969694312499794617172353154, 5.82927688263784182177139260274, 6.63365334605281608335065596385, 7.16144675812605530305332986431, 8.664363078139701017471117013203, 9.144677231113018676432330809714, 10.17795088602520798724835935767, 11.40667781835469141118610820852, 12.91520063742106788963729753229

Graph of the ZZ-function along the critical line