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Properties

Label 2-110-55.13-c1-0-3
Degree 22
Conductor 110110
Sign −0.228+0.973i-0.228 + 0.973i
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 + (−1.23 − 0.194i)3-s + (0.587 − 0.809i)4-s + (−1.87 − 1.22i)5-s + (1.18 − 0.384i)6-s + (−0.466 − 2.94i)7-s + (−0.156 + 0.987i)8-s + (−1.37 − 0.447i)9-s + (2.22 + 0.242i)10-s + (3.28 − 0.461i)11-s + (−0.880 + 0.880i)12-s + (−2.73 − 5.36i)13-s + (1.75 + 2.41i)14-s + (2.06 + 1.87i)15-s + (−0.309 − 0.951i)16-s + (−3.00 + 5.89i)17-s + ⋯
L(s)  = 1  + (−0.630 + 0.321i)2-s + (−0.710 − 0.112i)3-s + (0.293 − 0.404i)4-s + (−0.836 − 0.547i)5-s + (0.483 − 0.157i)6-s + (−0.176 − 1.11i)7-s + (−0.0553 + 0.349i)8-s + (−0.459 − 0.149i)9-s + (0.702 + 0.0765i)10-s + (0.990 − 0.139i)11-s + (−0.254 + 0.254i)12-s + (−0.757 − 1.48i)13-s + (0.468 + 0.645i)14-s + (0.532 + 0.483i)15-s + (−0.0772 − 0.237i)16-s + (−0.728 + 1.42i)17-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=((−0.228+0.973i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.228 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+1/2) L(s)=((−0.228+0.973i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: −0.228+0.973i-0.228 + 0.973i
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), −0.228+0.973i)(2,\ 110,\ (\ :1/2),\ -0.228 + 0.973i)

Particular Values

L(1)L(1) ≈\approx 0.247815−0.312672i0.247815 - 0.312672i
L(12)L(\frac12) ≈\approx 0.247815−0.312672i0.247815 - 0.312672i
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+(1.87+1.22i)T 1 + (1.87 + 1.22i)T
11 1+(−3.28+0.461i)T 1 + (-3.28 + 0.461i)T
good3 1+(1.23+0.194i)T+(2.85+0.927i)T2 1 + (1.23 + 0.194i)T + (2.85 + 0.927i)T^{2}
7 1+(0.466+2.94i)T+(−6.65+2.16i)T2 1 + (0.466 + 2.94i)T + (-6.65 + 2.16i)T^{2}
13 1+(2.73+5.36i)T+(−7.64+10.5i)T2 1 + (2.73 + 5.36i)T + (-7.64 + 10.5i)T^{2}
17 1+(3.00−5.89i)T+(−9.99−13.7i)T2 1 + (3.00 - 5.89i)T + (-9.99 - 13.7i)T^{2}
19 1+(−1.55+1.13i)T+(5.87−18.0i)T2 1 + (-1.55 + 1.13i)T + (5.87 - 18.0i)T^{2}
23 1+(−0.606−0.606i)T+23iT2 1 + (-0.606 - 0.606i)T + 23iT^{2}
29 1+(3.56+2.58i)T+(8.96+27.5i)T2 1 + (3.56 + 2.58i)T + (8.96 + 27.5i)T^{2}
31 1+(0.337−1.03i)T+(−25.0−18.2i)T2 1 + (0.337 - 1.03i)T + (-25.0 - 18.2i)T^{2}
37 1+(−7.26+1.15i)T+(35.1−11.4i)T2 1 + (-7.26 + 1.15i)T + (35.1 - 11.4i)T^{2}
41 1+(4.84+6.66i)T+(−12.6+38.9i)T2 1 + (4.84 + 6.66i)T + (-12.6 + 38.9i)T^{2}
43 1+(−3.90+3.90i)T−43iT2 1 + (-3.90 + 3.90i)T - 43iT^{2}
47 1+(−1.14+7.22i)T+(−44.6−14.5i)T2 1 + (-1.14 + 7.22i)T + (-44.6 - 14.5i)T^{2}
53 1+(−4.55+2.32i)T+(31.1−42.8i)T2 1 + (-4.55 + 2.32i)T + (31.1 - 42.8i)T^{2}
59 1+(−4.39+6.04i)T+(−18.2−56.1i)T2 1 + (-4.39 + 6.04i)T + (-18.2 - 56.1i)T^{2}
61 1+(4.11−1.33i)T+(49.3−35.8i)T2 1 + (4.11 - 1.33i)T + (49.3 - 35.8i)T^{2}
67 1+(1.06−1.06i)T−67iT2 1 + (1.06 - 1.06i)T - 67iT^{2}
71 1+(−1.18−3.65i)T+(−57.4+41.7i)T2 1 + (-1.18 - 3.65i)T + (-57.4 + 41.7i)T^{2}
73 1+(−0.684+0.108i)T+(69.4−22.5i)T2 1 + (-0.684 + 0.108i)T + (69.4 - 22.5i)T^{2}
79 1+(−1.26+3.88i)T+(−63.9−46.4i)T2 1 + (-1.26 + 3.88i)T + (-63.9 - 46.4i)T^{2}
83 1+(0.881+0.449i)T+(48.7+67.1i)T2 1 + (0.881 + 0.449i)T + (48.7 + 67.1i)T^{2}
89 1−12.5iT−89T2 1 - 12.5iT - 89T^{2}
97 1+(2.26+4.45i)T+(−57.0+78.4i)T2 1 + (2.26 + 4.45i)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

−13.20745450629782278180956098196, −12.16709273916606570464468286557, −11.17286544538601893017623672032, −10.33002967514491640881177414490, −8.916793204170908861944917586804, −7.84106964682532978085128569786, −6.79355118372691834440958054201, −5.46863230365923762869097884591, −3.86049662298231527757833314468, −0.59035972646171440629408156086, 2.67226787918358878276110199603, 4.56342406878667376965849257020, 6.29337483110922613490163499968, 7.30752988425534361748469980639, 8.841307529029690848019524160208, 9.596455841125042274255049900216, 11.28585589924687723062603437451, 11.58216952407677305854697326468, 12.29471399113750117118090761238, 14.13296733568280238133895993341

Graph of the ZZ-function along the critical line