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Properties

Label 2-110-11.4-c1-0-0
Degree 22
Conductor 110110
Sign −0.0320−0.999i-0.0320 - 0.999i
Analytic cond. 0.8783540.878354
Root an. cond. 0.9372050.937205
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  + (0.809 + 0.587i)2-s + (−0.650 + 2.00i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + (−1.70 + 1.23i)6-s + (−0.675 − 2.07i)7-s + (−0.309 + 0.951i)8-s + (−1.15 − 0.841i)9-s − 10-s + (1.92 − 2.69i)11-s − 2.10·12-s + (5.11 + 3.71i)13-s + (0.675 − 2.07i)14-s + (−0.650 − 2.00i)15-s + (−0.809 + 0.587i)16-s + (−1.34 + 0.980i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.375 + 1.15i)3-s + (0.154 + 0.475i)4-s + (−0.361 + 0.262i)5-s + (−0.695 + 0.505i)6-s + (−0.255 − 0.785i)7-s + (−0.109 + 0.336i)8-s + (−0.386 − 0.280i)9-s − 0.316·10-s + (0.581 − 0.813i)11-s − 0.607·12-s + (1.41 + 1.03i)13-s + (0.180 − 0.555i)14-s + (−0.167 − 0.516i)15-s + (−0.202 + 0.146i)16-s + (−0.327 + 0.237i)17-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=((−0.0320−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0320 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+1/2) L(s)=((−0.0320−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0320 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: −0.0320−0.999i-0.0320 - 0.999i
Analytic conductor: 0.8783540.878354
Root analytic conductor: 0.9372050.937205
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ110(81,⋅)\chi_{110} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 110, ( :1/2), −0.0320−0.999i)(2,\ 110,\ (\ :1/2),\ -0.0320 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.832391+0.859550i0.832391 + 0.859550i
L(12)L(\frac12) ≈\approx 0.832391+0.859550i0.832391 + 0.859550i
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+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
5 1+(0.809−0.587i)T 1 + (0.809 - 0.587i)T
11 1+(−1.92+2.69i)T 1 + (-1.92 + 2.69i)T
good3 1+(0.650−2.00i)T+(−2.42−1.76i)T2 1 + (0.650 - 2.00i)T + (-2.42 - 1.76i)T^{2}
7 1+(0.675+2.07i)T+(−5.66+4.11i)T2 1 + (0.675 + 2.07i)T + (-5.66 + 4.11i)T^{2}
13 1+(−5.11−3.71i)T+(4.01+12.3i)T2 1 + (-5.11 - 3.71i)T + (4.01 + 12.3i)T^{2}
17 1+(1.34−0.980i)T+(5.25−16.1i)T2 1 + (1.34 - 0.980i)T + (5.25 - 16.1i)T^{2}
19 1+(−1.82+5.60i)T+(−15.3−11.1i)T2 1 + (-1.82 + 5.60i)T + (-15.3 - 11.1i)T^{2}
23 1+3.26T+23T2 1 + 3.26T + 23T^{2}
29 1+(1.95+6.00i)T+(−23.4+17.0i)T2 1 + (1.95 + 6.00i)T + (-23.4 + 17.0i)T^{2}
31 1+(3.74+2.72i)T+(9.57+29.4i)T2 1 + (3.74 + 2.72i)T + (9.57 + 29.4i)T^{2}
37 1+(0.849+2.61i)T+(−29.9+21.7i)T2 1 + (0.849 + 2.61i)T + (-29.9 + 21.7i)T^{2}
41 1+(2.89−8.90i)T+(−33.1−24.0i)T2 1 + (2.89 - 8.90i)T + (-33.1 - 24.0i)T^{2}
43 1−1.29T+43T2 1 - 1.29T + 43T^{2}
47 1+(−0.582+1.79i)T+(−38.0−27.6i)T2 1 + (-0.582 + 1.79i)T + (-38.0 - 27.6i)T^{2}
53 1+(6.36+4.62i)T+(16.3+50.4i)T2 1 + (6.36 + 4.62i)T + (16.3 + 50.4i)T^{2}
59 1+(−2.47−7.63i)T+(−47.7+34.6i)T2 1 + (-2.47 - 7.63i)T + (-47.7 + 34.6i)T^{2}
61 1+(−3.51+2.55i)T+(18.8−58.0i)T2 1 + (-3.51 + 2.55i)T + (18.8 - 58.0i)T^{2}
67 1−5.61T+67T2 1 - 5.61T + 67T^{2}
71 1+(11.1−8.10i)T+(21.9−67.5i)T2 1 + (11.1 - 8.10i)T + (21.9 - 67.5i)T^{2}
73 1+(−4.32−13.3i)T+(−59.0+42.9i)T2 1 + (-4.32 - 13.3i)T + (-59.0 + 42.9i)T^{2}
79 1+(1.40+1.02i)T+(24.4+75.1i)T2 1 + (1.40 + 1.02i)T + (24.4 + 75.1i)T^{2}
83 1+(7.25−5.27i)T+(25.6−78.9i)T2 1 + (7.25 - 5.27i)T + (25.6 - 78.9i)T^{2}
89 1−13.3T+89T2 1 - 13.3T + 89T^{2}
97 1+(9.72+7.06i)T+(29.9+92.2i)T2 1 + (9.72 + 7.06i)T + (29.9 + 92.2i)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

−13.92606969781827734011197713882, −13.25924323600472505776993372078, −11.32445193223612458914362496832, −11.22861070096016187705338665510, −9.754541321505697330857018621409, −8.580069537788415620838678924642, −7.00559574022296404104842225870, −5.92668513505927428335398998632, −4.31216959498749642487471087521, −3.68825457008564817390724446662, 1.60723876779165351558593010346, 3.63049455671947928067562183886, 5.49954960020109966081034279669, 6.44825516578789026062001732864, 7.72996963386780191064911582078, 9.056881361958071134853068963183, 10.54789210899685806361513179030, 11.82059917985809652901718436237, 12.39325572033259848767265856262, 13.03235290782806362180391455517

Graph of the ZZ-function along the critical line