↑

Properties

Label 2-33e2-99.76-c0-0-1
Degree 22
Conductor 10891089
Sign −0.262+0.964i-0.262 + 0.964i
Analytic cond. 0.5434810.543481
Root an. cond. 0.7372120.737212
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 0.707i)2-s + (0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 1.41i·6-s + (−1.22 + 0.707i)7-s + (−0.499 − 0.866i)9-s − 1.41i·10-s + (−0.5 − 0.866i)12-s + (−0.999 + 1.73i)14-s + (−0.499 − 0.866i)15-s + (0.499 + 0.866i)16-s + (−1.22 − 0.707i)18-s + (−0.5 − 0.866i)20-s + 1.41i·21-s + ⋯
L(s)  = 1  + (1.22 − 0.707i)2-s + (0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 1.41i·6-s + (−1.22 + 0.707i)7-s + (−0.499 − 0.866i)9-s − 1.41i·10-s + (−0.5 − 0.866i)12-s + (−0.999 + 1.73i)14-s + (−0.499 − 0.866i)15-s + (0.499 + 0.866i)16-s + (−1.22 − 0.707i)18-s + (−0.5 − 0.866i)20-s + 1.41i·21-s + ⋯

Functional equation

Λ(s)=(1089s/2 ΓC(s) L(s)=((−0.262+0.964i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1089s/2 ΓC(s) L(s)=((−0.262+0.964i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10891089    =    32⋅1123^{2} \cdot 11^{2}
Sign: −0.262+0.964i-0.262 + 0.964i
Analytic conductor: 0.5434810.543481
Root analytic conductor: 0.7372120.737212
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1089(967,⋅)\chi_{1089} (967, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1089, ( :0), −0.262+0.964i)(2,\ 1089,\ (\ :0),\ -0.262 + 0.964i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.1038756572.103875657
L(12)L(\frac12) ≈\approx 2.1038756572.103875657
L(1)L(1) 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)
bad3 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
11 1 1
good2 1+(−1.22+0.707i)T+(0.5−0.866i)T2 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2}
5 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
7 1+(1.22−0.707i)T+(0.5−0.866i)T2 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}
13 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1−T2 1 - T^{2}
19 1−T2 1 - T^{2}
23 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
29 1+(−1.22+0.707i)T+(0.5−0.866i)T2 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2}
31 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
37 1+T+T2 1 + T + T^{2}
41 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
43 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
47 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
53 1−T+T2 1 - T + T^{2}
59 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
61 1+(1.22−0.707i)T+(0.5−0.866i)T2 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}
67 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
71 1+T+T2 1 + T + T^{2}
73 1+1.41iT−T2 1 + 1.41iT - T^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1+(1.22−0.707i)T+(0.5−0.866i)T2 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}
89 1+T2 1 + T^{2}
97 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{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

−9.770163954859646592040401083465, −8.966222470897175577346281033624, −8.415217593764334968797809938997, −7.10297103897251148556946399910, −6.10160206583250533886459593606, −5.60857862348732504522626856422, −4.50045355799031918187645100809, −3.29622586802518816694758198894, −2.65769916299205601557110762508, −1.53911232748449003147121250377, 2.66182498676302749541908147473, 3.42175591788946648275141168233, 4.10667743345344212108250949011, 5.14571203954136845081634654854, 6.05841395278505951749810063999, 6.76905103510706204498182617591, 7.41655155789646222772413484922, 8.688234422047237810482274753567, 9.748844464190782596013385895739, 10.17277726283853219649859807951

Graph of the ZZ-function along the critical line