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Properties

Label 2-110-5.4-c1-0-4
Degree 22
Conductor 110110
Sign −0.447+0.894i-0.447 + 0.894i
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  − i·2-s − i·3-s − 4-s + (−2 − i)5-s − 6-s − 3i·7-s + i·8-s + 2·9-s + (−1 + 2i)10-s + 11-s + i·12-s + 4i·13-s − 3·14-s + (−1 + 2i)15-s + 16-s − 3i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.894 − 0.447i)5-s − 0.408·6-s − 1.13i·7-s + 0.353i·8-s + 0.666·9-s + (−0.316 + 0.632i)10-s + 0.301·11-s + 0.288i·12-s + 1.10i·13-s − 0.801·14-s + (−0.258 + 0.516i)15-s + 0.250·16-s − 0.727i·17-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=((−0.447+0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+1/2) L(s)=((−0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: −0.447+0.894i-0.447 + 0.894i
Analytic conductor: 0.8783540.878354
Root analytic conductor: 0.9372050.937205
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ110(89,⋅)\chi_{110} (89, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 110, ( :1/2), −0.447+0.894i)(2,\ 110,\ (\ :1/2),\ -0.447 + 0.894i)

Particular Values

L(1)L(1) ≈\approx 0.475490−0.769359i0.475490 - 0.769359i
L(12)L(\frac12) ≈\approx 0.475490−0.769359i0.475490 - 0.769359i
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+iT 1 + iT
5 1+(2+i)T 1 + (2 + i)T
11 1−T 1 - T
good3 1+iT−3T2 1 + iT - 3T^{2}
7 1+3iT−7T2 1 + 3iT - 7T^{2}
13 1−4iT−13T2 1 - 4iT - 13T^{2}
17 1+3iT−17T2 1 + 3iT - 17T^{2}
19 1−5T+19T2 1 - 5T + 19T^{2}
23 1−4iT−23T2 1 - 4iT - 23T^{2}
29 1+5T+29T2 1 + 5T + 29T^{2}
31 1−7T+31T2 1 - 7T + 31T^{2}
37 1−7iT−37T2 1 - 7iT - 37T^{2}
41 1+8T+41T2 1 + 8T + 41T^{2}
43 1+6iT−43T2 1 + 6iT - 43T^{2}
47 1+8iT−47T2 1 + 8iT - 47T^{2}
53 1−9iT−53T2 1 - 9iT - 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+13T+61T2 1 + 13T + 61T^{2}
67 1−12iT−67T2 1 - 12iT - 67T^{2}
71 1+3T+71T2 1 + 3T + 71T^{2}
73 1+6iT−73T2 1 + 6iT - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1−4iT−83T2 1 - 4iT - 83T^{2}
89 1−15T+89T2 1 - 15T + 89T^{2}
97 1−12iT−97T2 1 - 12iT - 97T^{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

−13.45559146327761817173452832323, −11.99190237894301110032142643339, −11.64126003200319204649037279643, −10.23030438801077257222125559895, −9.153928025466994275930952749671, −7.69900240295976557573366582433, −6.96274046055113740750404610805, −4.73295666042138592083941452174, −3.67809253555074976153314280703, −1.23979701549011126191742279765, 3.34762356356885543961032986435, 4.76722448442396233580508980739, 6.09299157010974102633359664300, 7.48095748384322237013966681741, 8.465813242576305933897210105602, 9.651846457079291519356964694508, 10.75261710960811921867710425375, 12.03473824417203603620982899101, 12.90607822286981111208006585851, 14.45421972373257204925312642434

Graph of the ZZ-function along the critical line