↑

Properties

Label 2-110-11.3-c1-0-3
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

Related objects

Downloads

Learn more

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(91,⋅)\chi_{110} (91, \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}
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

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

Graph of the ZZ-function along the critical line