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Properties

Label 2-110-55.18-c1-0-4
Degree 22
Conductor 110110
Sign 0.567+0.823i0.567 + 0.823i
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  + (−0.156 − 0.987i)2-s + (1.48 − 0.757i)3-s + (−0.951 + 0.309i)4-s + (0.923 + 2.03i)5-s + (−0.980 − 1.35i)6-s + (1.49 − 2.92i)7-s + (0.453 + 0.891i)8-s + (−0.126 + 0.174i)9-s + (1.86 − 1.23i)10-s + (−3.16 − 0.977i)11-s + (−1.18 + 1.18i)12-s + (−6.10 + 0.966i)13-s + (−3.12 − 1.01i)14-s + (2.91 + 2.32i)15-s + (0.809 − 0.587i)16-s + (5.30 + 0.839i)17-s + ⋯
L(s)  = 1  + (−0.110 − 0.698i)2-s + (0.858 − 0.437i)3-s + (−0.475 + 0.154i)4-s + (0.413 + 0.910i)5-s + (−0.400 − 0.551i)6-s + (0.564 − 1.10i)7-s + (0.160 + 0.315i)8-s + (−0.0421 + 0.0580i)9-s + (0.590 − 0.389i)10-s + (−0.955 − 0.294i)11-s + (−0.340 + 0.340i)12-s + (−1.69 + 0.268i)13-s + (−0.835 − 0.271i)14-s + (0.753 + 0.601i)15-s + (0.202 − 0.146i)16-s + (1.28 + 0.203i)17-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=((0.567+0.823i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+1/2) L(s)=((0.567+0.823i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: 0.567+0.823i0.567 + 0.823i
Analytic conductor: 0.8783540.878354
Root analytic conductor: 0.9372050.937205
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ110(73,⋅)\chi_{110} (73, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 110, ( :1/2), 0.567+0.823i)(2,\ 110,\ (\ :1/2),\ 0.567 + 0.823i)

Particular Values

L(1)L(1) ≈\approx 1.07587−0.565399i1.07587 - 0.565399i
L(12)L(\frac12) ≈\approx 1.07587−0.565399i1.07587 - 0.565399i
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+(0.156+0.987i)T 1 + (0.156 + 0.987i)T
5 1+(−0.923−2.03i)T 1 + (-0.923 - 2.03i)T
11 1+(3.16+0.977i)T 1 + (3.16 + 0.977i)T
good3 1+(−1.48+0.757i)T+(1.76−2.42i)T2 1 + (-1.48 + 0.757i)T + (1.76 - 2.42i)T^{2}
7 1+(−1.49+2.92i)T+(−4.11−5.66i)T2 1 + (-1.49 + 2.92i)T + (-4.11 - 5.66i)T^{2}
13 1+(6.10−0.966i)T+(12.3−4.01i)T2 1 + (6.10 - 0.966i)T + (12.3 - 4.01i)T^{2}
17 1+(−5.30−0.839i)T+(16.1+5.25i)T2 1 + (-5.30 - 0.839i)T + (16.1 + 5.25i)T^{2}
19 1+(0.213−0.657i)T+(−15.3−11.1i)T2 1 + (0.213 - 0.657i)T + (-15.3 - 11.1i)T^{2}
23 1+(−3.92−3.92i)T+23iT2 1 + (-3.92 - 3.92i)T + 23iT^{2}
29 1+(0.689+2.12i)T+(−23.4+17.0i)T2 1 + (0.689 + 2.12i)T + (-23.4 + 17.0i)T^{2}
31 1+(4.63+3.37i)T+(9.57+29.4i)T2 1 + (4.63 + 3.37i)T + (9.57 + 29.4i)T^{2}
37 1+(2.91+1.48i)T+(21.7+29.9i)T2 1 + (2.91 + 1.48i)T + (21.7 + 29.9i)T^{2}
41 1+(−1.24−0.404i)T+(33.1+24.0i)T2 1 + (-1.24 - 0.404i)T + (33.1 + 24.0i)T^{2}
43 1+(−4.38+4.38i)T−43iT2 1 + (-4.38 + 4.38i)T - 43iT^{2}
47 1+(−0.800−1.57i)T+(−27.6+38.0i)T2 1 + (-0.800 - 1.57i)T + (-27.6 + 38.0i)T^{2}
53 1+(0.489+3.09i)T+(−50.4+16.3i)T2 1 + (0.489 + 3.09i)T + (-50.4 + 16.3i)T^{2}
59 1+(0.820−0.266i)T+(47.7−34.6i)T2 1 + (0.820 - 0.266i)T + (47.7 - 34.6i)T^{2}
61 1+(5.96+8.20i)T+(−18.8+58.0i)T2 1 + (5.96 + 8.20i)T + (-18.8 + 58.0i)T^{2}
67 1+(−9.05+9.05i)T−67iT2 1 + (-9.05 + 9.05i)T - 67iT^{2}
71 1+(−1.58+1.15i)T+(21.9−67.5i)T2 1 + (-1.58 + 1.15i)T + (21.9 - 67.5i)T^{2}
73 1+(2.99+1.52i)T+(42.9+59.0i)T2 1 + (2.99 + 1.52i)T + (42.9 + 59.0i)T^{2}
79 1+(−3.40−2.47i)T+(24.4+75.1i)T2 1 + (-3.40 - 2.47i)T + (24.4 + 75.1i)T^{2}
83 1+(−0.944+5.96i)T+(−78.9−25.6i)T2 1 + (-0.944 + 5.96i)T + (-78.9 - 25.6i)T^{2}
89 1−4.90iT−89T2 1 - 4.90iT - 89T^{2}
97 1+(−14.9+2.36i)T+(92.2−29.9i)T2 1 + (-14.9 + 2.36i)T + (92.2 - 29.9i)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

−13.68116675857975232651428867293, −12.62275645950030469651633521972, −11.19743473688075425516994344168, −10.39606553641156762551671369716, −9.460934471553127249226367763319, −7.73844560648048008237879335269, −7.44257599016247310979829358799, −5.23926807000446600239925951034, −3.37296137177898831375214993092, −2.14260139310860965127388052435, 2.63027466184134752954792097827, 4.85080199626155258351381644451, 5.53297526743254050854104585323, 7.53860060093696951787991338374, 8.515444928603924714252189184257, 9.274941194094353131768071934384, 10.15640467438504070231989626403, 12.13918155705355191110245255840, 12.80441849269761696036718872296, 14.27236438013607311341586640113

Graph of the ZZ-function along the critical line