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Properties

Label 2-110-5.4-c3-0-10
Degree 22
Conductor 110110
Sign −0.662+0.749i-0.662 + 0.749i
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  − 2i·2-s + 6.82i·3-s − 4·4-s + (−8.37 − 7.40i)5-s + 13.6·6-s − 13.6i·7-s + 8i·8-s − 19.5·9-s + (−14.8 + 16.7i)10-s − 11·11-s − 27.2i·12-s − 66.9i·13-s − 27.2·14-s + (50.5 − 57.1i)15-s + 16·16-s − 136. i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.31i·3-s − 0.5·4-s + (−0.749 − 0.662i)5-s + 0.928·6-s − 0.736i·7-s + 0.353i·8-s − 0.723·9-s + (−0.468 + 0.529i)10-s − 0.301·11-s − 0.656i·12-s − 1.42i·13-s − 0.520·14-s + (0.869 − 0.983i)15-s + 0.250·16-s − 1.95i·17-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=((−0.662+0.749i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+3/2) L(s)=((−0.662+0.749i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: −0.662+0.749i-0.662 + 0.749i
Analytic conductor: 6.490216.49021
Root analytic conductor: 2.547582.54758
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ110(89,⋅)\chi_{110} (89, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 110, ( :3/2), −0.662+0.749i)(2,\ 110,\ (\ :3/2),\ -0.662 + 0.749i)

Particular Values

L(2)L(2) ≈\approx 0.301836−0.669950i0.301836 - 0.669950i
L(12)L(\frac12) ≈\approx 0.301836−0.669950i0.301836 - 0.669950i
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+2iT 1 + 2iT
5 1+(8.37+7.40i)T 1 + (8.37 + 7.40i)T
11 1+11T 1 + 11T
good3 1−6.82iT−27T2 1 - 6.82iT - 27T^{2}
7 1+13.6iT−343T2 1 + 13.6iT - 343T^{2}
13 1+66.9iT−2.19e3T2 1 + 66.9iT - 2.19e3T^{2}
17 1+136.iT−4.91e3T2 1 + 136. iT - 4.91e3T^{2}
19 1+126.T+6.85e3T2 1 + 126.T + 6.85e3T^{2}
23 1−62.5iT−1.21e4T2 1 - 62.5iT - 1.21e4T^{2}
29 1+144.T+2.43e4T2 1 + 144.T + 2.43e4T^{2}
31 1−229.T+2.97e4T2 1 - 229.T + 2.97e4T^{2}
37 1+6.56iT−5.06e4T2 1 + 6.56iT - 5.06e4T^{2}
41 1+276.T+6.89e4T2 1 + 276.T + 6.89e4T^{2}
43 1+233.iT−7.95e4T2 1 + 233. iT - 7.95e4T^{2}
47 1−381.iT−1.03e5T2 1 - 381. iT - 1.03e5T^{2}
53 1+182.iT−1.48e5T2 1 + 182. iT - 1.48e5T^{2}
59 1+111.T+2.05e5T2 1 + 111.T + 2.05e5T^{2}
61 1−116.T+2.26e5T2 1 - 116.T + 2.26e5T^{2}
67 1+480.iT−3.00e5T2 1 + 480. iT - 3.00e5T^{2}
71 1+602.T+3.57e5T2 1 + 602.T + 3.57e5T^{2}
73 1−622.iT−3.89e5T2 1 - 622. iT - 3.89e5T^{2}
79 1−768.T+4.93e5T2 1 - 768.T + 4.93e5T^{2}
83 1+953.iT−5.71e5T2 1 + 953. iT - 5.71e5T^{2}
89 1−757.T+7.04e5T2 1 - 757.T + 7.04e5T^{2}
97 1+1.11e3iT−9.12e5T2 1 + 1.11e3iT - 9.12e5T^{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

−12.72226596640337241352028424867, −11.51189308958403259151014210833, −10.62888604118241817271227654951, −9.837154319220675796215232379867, −8.750746299560332971525630213845, −7.56006346583797721513148257808, −5.20231228329217200282836601158, −4.36417644245579532387535135971, −3.22392619478303466255590899575, −0.39742227369925742370582013549, 2.08353745027255629824845475999, 4.17055308184002242806594899116, 6.22606896849344642622255523480, 6.73523821168346963236774683911, 7.997941715485798357060330942670, 8.672266589184963061986179000429, 10.49858648181880787506225399030, 11.81509576950247482707799403539, 12.58293138604942079184315460737, 13.53319111451302412198105569292

Graph of the ZZ-function along the critical line