↑

Properties

Label 2-1110-37.10-c1-0-6
Degree 22
Conductor 11101110
Sign 0.729−0.683i0.729 - 0.683i
Analytic cond. 8.863398.86339
Root an. cond. 2.977142.97714
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + 0.999·6-s + (1.5 + 2.59i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s − 0.999·10-s − 2·11-s + (0.499 − 0.866i)12-s + (2.5 + 4.33i)13-s + 3·14-s + (0.499 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−1 + 1.73i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + 0.408·6-s + (0.566 + 0.981i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s − 0.316·10-s − 0.603·11-s + (0.144 − 0.249i)12-s + (0.693 + 1.20i)13-s + 0.801·14-s + (0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.242 + 0.420i)17-s + ⋯

Functional equation

Λ(s)=(1110s/2 ΓC(s) L(s)=((0.729−0.683i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1110s/2 ΓC(s+1/2) L(s)=((0.729−0.683i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11101110    =    2⋅3⋅5⋅372 \cdot 3 \cdot 5 \cdot 37
Sign: 0.729−0.683i0.729 - 0.683i
Analytic conductor: 8.863398.86339
Root analytic conductor: 2.977142.97714
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1110(121,⋅)\chi_{1110} (121, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1110, ( :1/2), 0.729−0.683i)(2,\ 1110,\ (\ :1/2),\ 0.729 - 0.683i)

Particular Values

L(1)L(1) ≈\approx 1.8719627561.871962756
L(12)L(\frac12) ≈\approx 1.8719627561.871962756
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
3 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
5 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
37 1+(0.5+6.06i)T 1 + (0.5 + 6.06i)T
good7 1+(−1.5−2.59i)T+(−3.5+6.06i)T2 1 + (-1.5 - 2.59i)T + (-3.5 + 6.06i)T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
13 1+(−2.5−4.33i)T+(−6.5+11.2i)T2 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2}
17 1+(1−1.73i)T+(−8.5−14.7i)T2 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2}
19 1+(−2.5−4.33i)T+(−9.5+16.4i)T2 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2}
23 1+8T+23T2 1 + 8T + 23T^{2}
29 1−T+29T2 1 - T + 29T^{2}
31 1−8T+31T2 1 - 8T + 31T^{2}
41 1+(−4−6.92i)T+(−20.5+35.5i)T2 1 + (-4 - 6.92i)T + (-20.5 + 35.5i)T^{2}
43 1+43T2 1 + 43T^{2}
47 1−6T+47T2 1 - 6T + 47T^{2}
53 1+(6−10.3i)T+(−26.5−45.8i)T2 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2}
59 1+(−4+6.92i)T+(−29.5−51.0i)T2 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2}
61 1+(−5−8.66i)T+(−30.5+52.8i)T2 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2}
67 1+(1+1.73i)T+(−33.5+58.0i)T2 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2}
71 1+(−0.5−0.866i)T+(−35.5+61.4i)T2 1 + (-0.5 - 0.866i)T + (-35.5 + 61.4i)T^{2}
73 1+73T2 1 + 73T^{2}
79 1+(2+3.46i)T+(−39.5+68.4i)T2 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2}
83 1+(−0.5+0.866i)T+(−41.5−71.8i)T2 1 + (-0.5 + 0.866i)T + (-41.5 - 71.8i)T^{2}
89 1+(−4+6.92i)T+(−44.5−77.0i)T2 1 + (-4 + 6.92i)T + (-44.5 - 77.0i)T^{2}
97 1+8T+97T2 1 + 8T + 97T^{2}
show more
show less
   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

−9.962461631172528544532136633171, −9.170583782538535838100764164529, −8.439372116882129520433634442255, −7.82308230809700285648835024222, −6.21037750487360919173461094893, −5.55751317488754659758590245493, −4.50194207772694664740192597835, −3.89801382029029491995489873891, −2.59423676419490241974096623353, −1.63577135326902415603914677464, 0.73359836761843542586620506714, 2.54594265830483708551903119224, 3.55765471115212399688304960601, 4.55324535471431792469270831168, 5.54917928030929152353684108601, 6.52719516192561633612746474034, 7.31977588978599633885090909763, 7.981503235449848832805616885141, 8.441415028354372868602428960285, 9.790350235133806941771929506128

Graph of the ZZ-function along the critical line