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Properties

Label 2-110-11.5-c3-0-4
Degree 22
Conductor 110110
Sign 0.915−0.401i0.915 - 0.401i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.618 − 1.90i)2-s + (5.32 + 3.86i)3-s + (−3.23 + 2.35i)4-s + (−1.54 + 4.75i)5-s + (4.06 − 12.5i)6-s + (7.43 − 5.40i)7-s + (6.47 + 4.70i)8-s + (5.04 + 15.5i)9-s + 10.0·10-s + (32.2 + 16.9i)11-s − 26.3·12-s + (24.6 + 75.7i)13-s + (−14.8 − 10.8i)14-s + (−26.6 + 19.3i)15-s + (4.94 − 15.2i)16-s + (−5.70 + 17.5i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (1.02 + 0.744i)3-s + (−0.404 + 0.293i)4-s + (−0.138 + 0.425i)5-s + (0.276 − 0.851i)6-s + (0.401 − 0.291i)7-s + (0.286 + 0.207i)8-s + (0.186 + 0.575i)9-s + 0.316·10-s + (0.885 + 0.465i)11-s − 0.633·12-s + (0.525 + 1.61i)13-s + (−0.284 − 0.206i)14-s + (−0.458 + 0.333i)15-s + (0.0772 − 0.237i)16-s + (−0.0813 + 0.250i)17-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=((0.915−0.401i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+3/2) L(s)=((0.915−0.401i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: 0.915−0.401i0.915 - 0.401i
Analytic conductor: 6.490216.49021
Root analytic conductor: 2.547582.54758
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ110(71,⋅)\chi_{110} (71, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 110, ( :3/2), 0.915−0.401i)(2,\ 110,\ (\ :3/2),\ 0.915 - 0.401i)

Particular Values

L(2)L(2) ≈\approx 1.90944+0.399816i1.90944 + 0.399816i
L(12)L(\frac12) ≈\approx 1.90944+0.399816i1.90944 + 0.399816i
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+(−32.2−16.9i)T 1 + (-32.2 - 16.9i)T
good3 1+(−5.32−3.86i)T+(8.34+25.6i)T2 1 + (-5.32 - 3.86i)T + (8.34 + 25.6i)T^{2}
7 1+(−7.43+5.40i)T+(105.−326.i)T2 1 + (-7.43 + 5.40i)T + (105. - 326. i)T^{2}
13 1+(−24.6−75.7i)T+(−1.77e3+1.29e3i)T2 1 + (-24.6 - 75.7i)T + (-1.77e3 + 1.29e3i)T^{2}
17 1+(5.70−17.5i)T+(−3.97e3−2.88e3i)T2 1 + (5.70 - 17.5i)T + (-3.97e3 - 2.88e3i)T^{2}
19 1+(−12.9−9.41i)T+(2.11e3+6.52e3i)T2 1 + (-12.9 - 9.41i)T + (2.11e3 + 6.52e3i)T^{2}
23 1−7.44T+1.21e4T2 1 - 7.44T + 1.21e4T^{2}
29 1+(−44.2+32.1i)T+(7.53e3−2.31e4i)T2 1 + (-44.2 + 32.1i)T + (7.53e3 - 2.31e4i)T^{2}
31 1+(55.5+170.i)T+(−2.41e4+1.75e4i)T2 1 + (55.5 + 170. i)T + (-2.41e4 + 1.75e4i)T^{2}
37 1+(−133.+96.7i)T+(1.56e4−4.81e4i)T2 1 + (-133. + 96.7i)T + (1.56e4 - 4.81e4i)T^{2}
41 1+(179.+130.i)T+(2.12e4+6.55e4i)T2 1 + (179. + 130. i)T + (2.12e4 + 6.55e4i)T^{2}
43 1+115.T+7.95e4T2 1 + 115.T + 7.95e4T^{2}
47 1+(380.+276.i)T+(3.20e4+9.87e4i)T2 1 + (380. + 276. i)T + (3.20e4 + 9.87e4i)T^{2}
53 1+(224.+691.i)T+(−1.20e5+8.75e4i)T2 1 + (224. + 691. i)T + (-1.20e5 + 8.75e4i)T^{2}
59 1+(114.−83.1i)T+(6.34e4−1.95e5i)T2 1 + (114. - 83.1i)T + (6.34e4 - 1.95e5i)T^{2}
61 1+(191.−590.i)T+(−1.83e5−1.33e5i)T2 1 + (191. - 590. i)T + (-1.83e5 - 1.33e5i)T^{2}
67 1−783.T+3.00e5T2 1 - 783.T + 3.00e5T^{2}
71 1+(99.8−307.i)T+(−2.89e5−2.10e5i)T2 1 + (99.8 - 307. i)T + (-2.89e5 - 2.10e5i)T^{2}
73 1+(−806.+586.i)T+(1.20e5−3.69e5i)T2 1 + (-806. + 586. i)T + (1.20e5 - 3.69e5i)T^{2}
79 1+(−54.1−166.i)T+(−3.98e5+2.89e5i)T2 1 + (-54.1 - 166. i)T + (-3.98e5 + 2.89e5i)T^{2}
83 1+(−214.+659.i)T+(−4.62e5−3.36e5i)T2 1 + (-214. + 659. i)T + (-4.62e5 - 3.36e5i)T^{2}
89 1+101.T+7.04e5T2 1 + 101.T + 7.04e5T^{2}
97 1+(−16.1−49.6i)T+(−7.38e5+5.36e5i)T2 1 + (-16.1 - 49.6i)T + (-7.38e5 + 5.36e5i)T^{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

−13.47881450648250839791448385984, −11.91374802976151544669028208453, −11.13323259048964494915362248490, −9.863337218339843916041457557804, −9.178836339133585218344912050477, −8.167067312005438211390957464408, −6.70706727599015435209827724511, −4.38498764097170279568610455069, −3.61806282192386613066974159009, −1.93252453882285229670585470730, 1.23185427506815647531564093111, 3.23650743115899426341393550059, 5.15462465326309848761268680779, 6.57012918759035079529729605687, 7.946636753421089880194442560691, 8.392619254499830148854726436639, 9.416407151085666820401739923210, 10.99196196346646940016724184708, 12.42705799078598520418494248061, 13.33156520281787727653218881462

Graph of the ZZ-function along the critical line