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Properties

Label 2-33e2-11.6-c0-0-0
Degree 22
Conductor 10891089
Sign −0.0938−0.995i-0.0938 - 0.995i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 + 0.587i)4-s + (0.831 + 1.14i)7-s + (−1.34 + 0.437i)13-s + (0.309 − 0.951i)16-s + (−0.831 + 1.14i)19-s + (0.809 + 0.587i)25-s + (−1.34 − 0.437i)28-s + 1.41i·43-s + (−0.309 + 0.951i)49-s + (0.831 − 1.14i)52-s + (1.34 + 0.437i)61-s + (0.309 + 0.951i)64-s + (−0.831 − 1.14i)73-s − 1.41i·76-s + (1.34 − 0.437i)79-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)4-s + (0.831 + 1.14i)7-s + (−1.34 + 0.437i)13-s + (0.309 − 0.951i)16-s + (−0.831 + 1.14i)19-s + (0.809 + 0.587i)25-s + (−1.34 − 0.437i)28-s + 1.41i·43-s + (−0.309 + 0.951i)49-s + (0.831 − 1.14i)52-s + (1.34 + 0.437i)61-s + (0.309 + 0.951i)64-s + (−0.831 − 1.14i)73-s − 1.41i·76-s + (1.34 − 0.437i)79-s + ⋯

Functional equation

Λ(s)=(1089s/2 ΓC(s) L(s)=((−0.0938−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0938 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1089s/2 ΓC(s) L(s)=((−0.0938−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0938 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10891089    =    32⋅1123^{2} \cdot 11^{2}
Sign: −0.0938−0.995i-0.0938 - 0.995i
Analytic conductor: 0.5434810.543481
Root analytic conductor: 0.7372120.737212
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1089(820,⋅)\chi_{1089} (820, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1089, ( :0), −0.0938−0.995i)(2,\ 1089,\ (\ :0),\ -0.0938 - 0.995i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.79771449540.7977144954
L(12)L(\frac12) ≈\approx 0.79771449540.7977144954
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 1
11 1 1
good2 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
5 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
7 1+(−0.831−1.14i)T+(−0.309+0.951i)T2 1 + (-0.831 - 1.14i)T + (-0.309 + 0.951i)T^{2}
13 1+(1.34−0.437i)T+(0.809−0.587i)T2 1 + (1.34 - 0.437i)T + (0.809 - 0.587i)T^{2}
17 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
19 1+(0.831−1.14i)T+(−0.309−0.951i)T2 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2}
23 1+T2 1 + T^{2}
29 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
31 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
37 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
41 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
43 1−1.41iT−T2 1 - 1.41iT - T^{2}
47 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
53 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
59 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
61 1+(−1.34−0.437i)T+(0.809+0.587i)T2 1 + (-1.34 - 0.437i)T + (0.809 + 0.587i)T^{2}
67 1+T2 1 + T^{2}
71 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
73 1+(0.831+1.14i)T+(−0.309+0.951i)T2 1 + (0.831 + 1.14i)T + (-0.309 + 0.951i)T^{2}
79 1+(−1.34+0.437i)T+(0.809−0.587i)T2 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2}
83 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
89 1+T2 1 + T^{2}
97 1+(0.618+1.90i)T+(−0.809+0.587i)T2 1 + (0.618 + 1.90i)T + (-0.809 + 0.587i)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

−10.08378027266777955122824549369, −9.335315909755807050048303636413, −8.607123413207573461843242586894, −8.009256356972757416304581333066, −7.14052110300060959759244997026, −5.87022071246022123282800388836, −4.99048785404188417271629906794, −4.36092048291769080420979355836, −3.05361966177090638218079065431, −1.94848483513618685276119927009, 0.76198671179119282246770457432, 2.31917304404941302629406842304, 3.90734582682606064973369848354, 4.74484380664329132083583427079, 5.22346012892849327190364256642, 6.58894840832320962476003324715, 7.37456768357368292289237611418, 8.265057042609235963932222246621, 9.030564040896637976408327841118, 10.00287834441950506099861226647

Graph of the ZZ-function along the critical line