↑

Properties

Label 2-110-11.9-c3-0-5
Degree 22
Conductor 110110
Sign 0.935+0.352i0.935 + 0.352i
Analytic cond. 6.490216.49021
Root an. cond. 2.547582.54758
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.618 + 1.90i)2-s + (−5.73 + 4.16i)3-s + (−3.23 − 2.35i)4-s + (−1.54 − 4.75i)5-s + (−4.38 − 13.4i)6-s + (−9.28 − 6.74i)7-s + (6.47 − 4.70i)8-s + (7.20 − 22.1i)9-s + 10.0·10-s + (31.3 − 18.6i)11-s + 28.3·12-s + (−11.2 + 34.4i)13-s + (18.5 − 13.4i)14-s + (28.6 + 20.8i)15-s + (4.94 + 15.2i)16-s + (−7.66 − 23.5i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−1.10 + 0.802i)3-s + (−0.404 − 0.293i)4-s + (−0.138 − 0.425i)5-s + (−0.298 − 0.917i)6-s + (−0.501 − 0.364i)7-s + (0.286 − 0.207i)8-s + (0.266 − 0.820i)9-s + 0.316·10-s + (0.859 − 0.511i)11-s + 0.682·12-s + (−0.239 + 0.736i)13-s + (0.354 − 0.257i)14-s + (0.493 + 0.358i)15-s + (0.0772 + 0.237i)16-s + (−0.109 − 0.336i)17-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=((0.935+0.352i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+3/2) L(s)=((0.935+0.352i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: 0.935+0.352i0.935 + 0.352i
Analytic conductor: 6.490216.49021
Root analytic conductor: 2.547582.54758
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ110(31,⋅)\chi_{110} (31, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 110, ( :3/2), 0.935+0.352i)(2,\ 110,\ (\ :3/2),\ 0.935 + 0.352i)

Particular Values

L(2)L(2) ≈\approx 0.619544−0.112747i0.619544 - 0.112747i
L(12)L(\frac12) ≈\approx 0.619544−0.112747i0.619544 - 0.112747i
L(52)L(\frac{5}{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+(0.618−1.90i)T 1 + (0.618 - 1.90i)T
5 1+(1.54+4.75i)T 1 + (1.54 + 4.75i)T
11 1+(−31.3+18.6i)T 1 + (-31.3 + 18.6i)T
good3 1+(5.73−4.16i)T+(8.34−25.6i)T2 1 + (5.73 - 4.16i)T + (8.34 - 25.6i)T^{2}
7 1+(9.28+6.74i)T+(105.+326.i)T2 1 + (9.28 + 6.74i)T + (105. + 326. i)T^{2}
13 1+(11.2−34.4i)T+(−1.77e3−1.29e3i)T2 1 + (11.2 - 34.4i)T + (-1.77e3 - 1.29e3i)T^{2}
17 1+(7.66+23.5i)T+(−3.97e3+2.88e3i)T2 1 + (7.66 + 23.5i)T + (-3.97e3 + 2.88e3i)T^{2}
19 1+(−93.8+68.2i)T+(2.11e3−6.52e3i)T2 1 + (-93.8 + 68.2i)T + (2.11e3 - 6.52e3i)T^{2}
23 1−89.1T+1.21e4T2 1 - 89.1T + 1.21e4T^{2}
29 1+(160.+116.i)T+(7.53e3+2.31e4i)T2 1 + (160. + 116. i)T + (7.53e3 + 2.31e4i)T^{2}
31 1+(−73.7+227.i)T+(−2.41e4−1.75e4i)T2 1 + (-73.7 + 227. i)T + (-2.41e4 - 1.75e4i)T^{2}
37 1+(251.+183.i)T+(1.56e4+4.81e4i)T2 1 + (251. + 183. i)T + (1.56e4 + 4.81e4i)T^{2}
41 1+(−30.1+21.9i)T+(2.12e4−6.55e4i)T2 1 + (-30.1 + 21.9i)T + (2.12e4 - 6.55e4i)T^{2}
43 1−330.T+7.95e4T2 1 - 330.T + 7.95e4T^{2}
47 1+(127.−92.6i)T+(3.20e4−9.87e4i)T2 1 + (127. - 92.6i)T + (3.20e4 - 9.87e4i)T^{2}
53 1+(11.4−35.1i)T+(−1.20e5−8.75e4i)T2 1 + (11.4 - 35.1i)T + (-1.20e5 - 8.75e4i)T^{2}
59 1+(516.+374.i)T+(6.34e4+1.95e5i)T2 1 + (516. + 374. i)T + (6.34e4 + 1.95e5i)T^{2}
61 1+(−10.3−31.9i)T+(−1.83e5+1.33e5i)T2 1 + (-10.3 - 31.9i)T + (-1.83e5 + 1.33e5i)T^{2}
67 1−396.T+3.00e5T2 1 - 396.T + 3.00e5T^{2}
71 1+(−337.−1.03e3i)T+(−2.89e5+2.10e5i)T2 1 + (-337. - 1.03e3i)T + (-2.89e5 + 2.10e5i)T^{2}
73 1+(276.+200.i)T+(1.20e5+3.69e5i)T2 1 + (276. + 200. i)T + (1.20e5 + 3.69e5i)T^{2}
79 1+(−113.+349.i)T+(−3.98e5−2.89e5i)T2 1 + (-113. + 349. i)T + (-3.98e5 - 2.89e5i)T^{2}
83 1+(288.+886.i)T+(−4.62e5+3.36e5i)T2 1 + (288. + 886. i)T + (-4.62e5 + 3.36e5i)T^{2}
89 1−470.T+7.04e5T2 1 - 470.T + 7.04e5T^{2}
97 1+(−361.+1.11e3i)T+(−7.38e5−5.36e5i)T2 1 + (-361. + 1.11e3i)T + (-7.38e5 - 5.36e5i)T^{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

−13.22916289950513462035698694691, −11.77340461011525610430411824248, −11.09532629467355708661120862733, −9.709862162236827819556618808523, −9.100605560471018967249881350028, −7.35876854947189900812560722377, −6.21473174205530879569662509421, −5.10823137546310638616315917691, −3.98650801381844387643017805984, −0.48339145173196210335635633668, 1.33188105394653926113455420905, 3.30983300613706949801940015186, 5.27962924304850510874383486159, 6.52478754600541840207374100160, 7.53206821486422474466772219773, 9.146287468043217912348693453204, 10.32511559913798792544067734141, 11.30881748487506115795753143006, 12.27523719879340623559957927622, 12.63788362956822254909359359502

Graph of the ZZ-function along the critical line