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Properties

Label 2-110-11.3-c3-0-6
Degree 22
Conductor 110110
Sign 0.738+0.674i0.738 + 0.674i
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  + (1.61 − 1.17i)2-s + (0.292 + 0.901i)3-s + (1.23 − 3.80i)4-s + (4.04 + 2.93i)5-s + (1.53 + 1.11i)6-s + (2.73 − 8.41i)7-s + (−2.47 − 7.60i)8-s + (21.1 − 15.3i)9-s + 10·10-s + (36.3 + 3.21i)11-s + 3.79·12-s + (2.56 − 1.86i)13-s + (−5.46 − 16.8i)14-s + (−1.46 + 4.50i)15-s + (−12.9 − 9.40i)16-s + (16.7 + 12.1i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.0563 + 0.173i)3-s + (0.154 − 0.475i)4-s + (0.361 + 0.262i)5-s + (0.104 + 0.0757i)6-s + (0.147 − 0.454i)7-s + (−0.109 − 0.336i)8-s + (0.782 − 0.568i)9-s + 0.316·10-s + (0.996 + 0.0881i)11-s + 0.0911·12-s + (0.0546 − 0.0397i)13-s + (−0.104 − 0.321i)14-s + (−0.0252 + 0.0775i)15-s + (−0.202 − 0.146i)16-s + (0.239 + 0.173i)17-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=((0.738+0.674i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+3/2) L(s)=((0.738+0.674i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: 0.738+0.674i0.738 + 0.674i
Analytic conductor: 6.490216.49021
Root analytic conductor: 2.547582.54758
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ110(91,⋅)\chi_{110} (91, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 110, ( :3/2), 0.738+0.674i)(2,\ 110,\ (\ :3/2),\ 0.738 + 0.674i)

Particular Values

L(2)L(2) ≈\approx 2.30080−0.892840i2.30080 - 0.892840i
L(12)L(\frac12) ≈\approx 2.30080−0.892840i2.30080 - 0.892840i
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+(−1.61+1.17i)T 1 + (-1.61 + 1.17i)T
5 1+(−4.04−2.93i)T 1 + (-4.04 - 2.93i)T
11 1+(−36.3−3.21i)T 1 + (-36.3 - 3.21i)T
good3 1+(−0.292−0.901i)T+(−21.8+15.8i)T2 1 + (-0.292 - 0.901i)T + (-21.8 + 15.8i)T^{2}
7 1+(−2.73+8.41i)T+(−277.−201.i)T2 1 + (-2.73 + 8.41i)T + (-277. - 201. i)T^{2}
13 1+(−2.56+1.86i)T+(678.−2.08e3i)T2 1 + (-2.56 + 1.86i)T + (678. - 2.08e3i)T^{2}
17 1+(−16.7−12.1i)T+(1.51e3+4.67e3i)T2 1 + (-16.7 - 12.1i)T + (1.51e3 + 4.67e3i)T^{2}
19 1+(14.3+44.2i)T+(−5.54e3+4.03e3i)T2 1 + (14.3 + 44.2i)T + (-5.54e3 + 4.03e3i)T^{2}
23 1+43.9T+1.21e4T2 1 + 43.9T + 1.21e4T^{2}
29 1+(13.0−40.0i)T+(−1.97e4−1.43e4i)T2 1 + (13.0 - 40.0i)T + (-1.97e4 - 1.43e4i)T^{2}
31 1+(177.−129.i)T+(9.20e3−2.83e4i)T2 1 + (177. - 129. i)T + (9.20e3 - 2.83e4i)T^{2}
37 1+(18.4−56.8i)T+(−4.09e4−2.97e4i)T2 1 + (18.4 - 56.8i)T + (-4.09e4 - 2.97e4i)T^{2}
41 1+(−22.2−68.5i)T+(−5.57e4+4.05e4i)T2 1 + (-22.2 - 68.5i)T + (-5.57e4 + 4.05e4i)T^{2}
43 1+368.T+7.95e4T2 1 + 368.T + 7.95e4T^{2}
47 1+(−83.7−257.i)T+(−8.39e4+6.10e4i)T2 1 + (-83.7 - 257. i)T + (-8.39e4 + 6.10e4i)T^{2}
53 1+(460.−334.i)T+(4.60e4−1.41e5i)T2 1 + (460. - 334. i)T + (4.60e4 - 1.41e5i)T^{2}
59 1+(−41.7+128.i)T+(−1.66e5−1.20e5i)T2 1 + (-41.7 + 128. i)T + (-1.66e5 - 1.20e5i)T^{2}
61 1+(61.3+44.5i)T+(7.01e4+2.15e5i)T2 1 + (61.3 + 44.5i)T + (7.01e4 + 2.15e5i)T^{2}
67 1−113.T+3.00e5T2 1 - 113.T + 3.00e5T^{2}
71 1+(458.+333.i)T+(1.10e5+3.40e5i)T2 1 + (458. + 333. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(215.−661.i)T+(−3.14e5−2.28e5i)T2 1 + (215. - 661. i)T + (-3.14e5 - 2.28e5i)T^{2}
79 1+(−590.+428.i)T+(1.52e5−4.68e5i)T2 1 + (-590. + 428. i)T + (1.52e5 - 4.68e5i)T^{2}
83 1+(924.+671.i)T+(1.76e5+5.43e5i)T2 1 + (924. + 671. i)T + (1.76e5 + 5.43e5i)T^{2}
89 1−1.08e3T+7.04e5T2 1 - 1.08e3T + 7.04e5T^{2}
97 1+(−186.+135.i)T+(2.82e5−8.68e5i)T2 1 + (-186. + 135. i)T + (2.82e5 - 8.68e5i)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

−13.03492399378651129323639061788, −12.11482171655512763353171240966, −10.97955991924758293313335293073, −10.01486204400581518037910477213, −9.050860902788494435512807669128, −7.22015598294542195166866029520, −6.21111910319215405663270185075, −4.59554029092388674146781058612, −3.44839032844192127786490174052, −1.46364707852849424203940481894, 1.89169770517639109155226621848, 3.91720347718470053841834797362, 5.26873521580286128125988650321, 6.44729556227921941692255667658, 7.66270566153220231078969519544, 8.857012785562844139846156790905, 10.06060843938190990550091767720, 11.50989739757048926926578388379, 12.45645143323966976279389075854, 13.36208250479887831870688617184

Graph of the ZZ-function along the critical line