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Properties

Label 2-110-55.13-c1-0-1
Degree 22
Conductor 110110
Sign 0.993−0.117i0.993 - 0.117i
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  + (−0.891 + 0.453i)2-s + (2.03 + 0.322i)3-s + (0.587 − 0.809i)4-s + (0.667 − 2.13i)5-s + (−1.95 + 0.636i)6-s + (−0.0611 − 0.386i)7-s + (−0.156 + 0.987i)8-s + (1.18 + 0.384i)9-s + (0.373 + 2.20i)10-s + (−3.24 + 0.696i)11-s + (1.45 − 1.45i)12-s + (2.60 + 5.11i)13-s + (0.229 + 0.316i)14-s + (2.04 − 4.12i)15-s + (−0.309 − 0.951i)16-s + (0.224 − 0.440i)17-s + ⋯
L(s)  = 1  + (−0.630 + 0.321i)2-s + (1.17 + 0.186i)3-s + (0.293 − 0.404i)4-s + (0.298 − 0.954i)5-s + (−0.799 + 0.259i)6-s + (−0.0231 − 0.145i)7-s + (−0.0553 + 0.349i)8-s + (0.394 + 0.128i)9-s + (0.118 + 0.697i)10-s + (−0.977 + 0.210i)11-s + (0.420 − 0.420i)12-s + (0.723 + 1.41i)13-s + (0.0614 + 0.0845i)14-s + (0.528 − 1.06i)15-s + (−0.0772 − 0.237i)16-s + (0.0544 − 0.106i)17-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=((0.993−0.117i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+1/2) L(s)=((0.993−0.117i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: 0.993−0.117i0.993 - 0.117i
Analytic conductor: 0.8783540.878354
Root analytic conductor: 0.9372050.937205
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ110(13,⋅)\chi_{110} (13, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 110, ( :1/2), 0.993−0.117i)(2,\ 110,\ (\ :1/2),\ 0.993 - 0.117i)

Particular Values

L(1)L(1) ≈\approx 1.07938+0.0638108i1.07938 + 0.0638108i
L(12)L(\frac12) ≈\approx 1.07938+0.0638108i1.07938 + 0.0638108i
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+(0.891−0.453i)T 1 + (0.891 - 0.453i)T
5 1+(−0.667+2.13i)T 1 + (-0.667 + 2.13i)T
11 1+(3.24−0.696i)T 1 + (3.24 - 0.696i)T
good3 1+(−2.03−0.322i)T+(2.85+0.927i)T2 1 + (-2.03 - 0.322i)T + (2.85 + 0.927i)T^{2}
7 1+(0.0611+0.386i)T+(−6.65+2.16i)T2 1 + (0.0611 + 0.386i)T + (-6.65 + 2.16i)T^{2}
13 1+(−2.60−5.11i)T+(−7.64+10.5i)T2 1 + (-2.60 - 5.11i)T + (-7.64 + 10.5i)T^{2}
17 1+(−0.224+0.440i)T+(−9.99−13.7i)T2 1 + (-0.224 + 0.440i)T + (-9.99 - 13.7i)T^{2}
19 1+(2.61−1.90i)T+(5.87−18.0i)T2 1 + (2.61 - 1.90i)T + (5.87 - 18.0i)T^{2}
23 1+(−1.05−1.05i)T+23iT2 1 + (-1.05 - 1.05i)T + 23iT^{2}
29 1+(6.85+4.97i)T+(8.96+27.5i)T2 1 + (6.85 + 4.97i)T + (8.96 + 27.5i)T^{2}
31 1+(1.58−4.87i)T+(−25.0−18.2i)T2 1 + (1.58 - 4.87i)T + (-25.0 - 18.2i)T^{2}
37 1+(−9.20+1.45i)T+(35.1−11.4i)T2 1 + (-9.20 + 1.45i)T + (35.1 - 11.4i)T^{2}
41 1+(3.50+4.82i)T+(−12.6+38.9i)T2 1 + (3.50 + 4.82i)T + (-12.6 + 38.9i)T^{2}
43 1+(−3.86+3.86i)T−43iT2 1 + (-3.86 + 3.86i)T - 43iT^{2}
47 1+(0.596−3.76i)T+(−44.6−14.5i)T2 1 + (0.596 - 3.76i)T + (-44.6 - 14.5i)T^{2}
53 1+(−1.62+0.830i)T+(31.1−42.8i)T2 1 + (-1.62 + 0.830i)T + (31.1 - 42.8i)T^{2}
59 1+(−1.77+2.44i)T+(−18.2−56.1i)T2 1 + (-1.77 + 2.44i)T + (-18.2 - 56.1i)T^{2}
61 1+(−7.86+2.55i)T+(49.3−35.8i)T2 1 + (-7.86 + 2.55i)T + (49.3 - 35.8i)T^{2}
67 1+(10.6−10.6i)T−67iT2 1 + (10.6 - 10.6i)T - 67iT^{2}
71 1+(−3.83−11.8i)T+(−57.4+41.7i)T2 1 + (-3.83 - 11.8i)T + (-57.4 + 41.7i)T^{2}
73 1+(11.9−1.89i)T+(69.4−22.5i)T2 1 + (11.9 - 1.89i)T + (69.4 - 22.5i)T^{2}
79 1+(−2.96+9.12i)T+(−63.9−46.4i)T2 1 + (-2.96 + 9.12i)T + (-63.9 - 46.4i)T^{2}
83 1+(−5.29−2.69i)T+(48.7+67.1i)T2 1 + (-5.29 - 2.69i)T + (48.7 + 67.1i)T^{2}
89 1+9.83iT−89T2 1 + 9.83iT - 89T^{2}
97 1+(2.44+4.79i)T+(−57.0+78.4i)T2 1 + (2.44 + 4.79i)T + (-57.0 + 78.4i)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

−13.77437361568574045183165830826, −12.97135781593774928720994717738, −11.48610002713363460454386284923, −10.07639961443334041229606356466, −9.115742336043115423668713200075, −8.523228528158739901560571248815, −7.43960283378527250784254966547, −5.78051010331515263249155063046, −4.12617195148352063365904496021, −2.08148055757081211289421928843, 2.44260292204020476882042747305, 3.34113885284566105267708371976, 5.88026588475879062335899655039, 7.48114539418584403252172092549, 8.205262721562147466666551457282, 9.303538530284405104591345527125, 10.47987068475535631605964019492, 11.15538681681899291569113915461, 12.97716872138964401332882186835, 13.43952276906345550151654621783

Graph of the ZZ-function along the critical line