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Properties

Label 2-33e2-1.1-c3-0-115
Degree 22
Conductor 10891089
Sign −1-1
Analytic cond. 64.253064.2530
Root an. cond. 8.015808.01580
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 4-s + 12·5-s − 12·7-s − 21·8-s + 36·10-s + 66·13-s − 36·14-s − 71·16-s − 114·17-s − 42·19-s + 12·20-s − 18·23-s + 19·25-s + 198·26-s − 12·28-s + 186·29-s − 308·31-s − 45·32-s − 342·34-s − 144·35-s − 146·37-s − 126·38-s − 252·40-s + 42·41-s + 366·43-s − 54·46-s + ⋯
L(s)  = 1  + 1.06·2-s + 1/8·4-s + 1.07·5-s − 0.647·7-s − 0.928·8-s + 1.13·10-s + 1.40·13-s − 0.687·14-s − 1.10·16-s − 1.62·17-s − 0.507·19-s + 0.134·20-s − 0.163·23-s + 0.151·25-s + 1.49·26-s − 0.0809·28-s + 1.19·29-s − 1.78·31-s − 0.248·32-s − 1.72·34-s − 0.695·35-s − 0.648·37-s − 0.537·38-s − 0.996·40-s + 0.159·41-s + 1.29·43-s − 0.173·46-s + ⋯

Functional equation

Λ(s)=(1089s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1089s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10891089    =    32⋅1123^{2} \cdot 11^{2}
Sign: −1-1
Analytic conductor: 64.253064.2530
Root analytic conductor: 8.015808.01580
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1089, ( :3/2), −1)(2,\ 1089,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad3 1 1
11 1 1
good2 1−3T+p3T2 1 - 3 T + p^{3} T^{2}
5 1−12T+p3T2 1 - 12 T + p^{3} T^{2}
7 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
13 1−66T+p3T2 1 - 66 T + p^{3} T^{2}
17 1+114T+p3T2 1 + 114 T + p^{3} T^{2}
19 1+42T+p3T2 1 + 42 T + p^{3} T^{2}
23 1+18T+p3T2 1 + 18 T + p^{3} T^{2}
29 1−186T+p3T2 1 - 186 T + p^{3} T^{2}
31 1+308T+p3T2 1 + 308 T + p^{3} T^{2}
37 1+146T+p3T2 1 + 146 T + p^{3} T^{2}
41 1−42T+p3T2 1 - 42 T + p^{3} T^{2}
43 1−366T+p3T2 1 - 366 T + p^{3} T^{2}
47 1+618T+p3T2 1 + 618 T + p^{3} T^{2}
53 1−408T+p3T2 1 - 408 T + p^{3} T^{2}
59 1−132T+p3T2 1 - 132 T + p^{3} T^{2}
61 1+630T+p3T2 1 + 630 T + p^{3} T^{2}
67 1+452T+p3T2 1 + 452 T + p^{3} T^{2}
71 1−282T+p3T2 1 - 282 T + p^{3} T^{2}
73 1+684T+p3T2 1 + 684 T + p^{3} T^{2}
79 1+1272T+p3T2 1 + 1272 T + p^{3} T^{2}
83 1+432T+p3T2 1 + 432 T + p^{3} T^{2}
89 1+954T+p3T2 1 + 954 T + p^{3} T^{2}
97 1−326T+p3T2 1 - 326 T + p^{3} 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

−8.994843914743951221101935483288, −8.596356318810173680753919017784, −6.94427527513079405464318448646, −6.18316284092337417489211352823, −5.80173663768574755323855184176, −4.66854448928458628169696553925, −3.83541396049379968015384410688, −2.84850810699099482663415166216, −1.74655279970249729963429530969, 0, 1.74655279970249729963429530969, 2.84850810699099482663415166216, 3.83541396049379968015384410688, 4.66854448928458628169696553925, 5.80173663768574755323855184176, 6.18316284092337417489211352823, 6.94427527513079405464318448646, 8.596356318810173680753919017784, 8.994843914743951221101935483288

Graph of the ZZ-function along the critical line