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Properties

Label 2-3276-91.38-c0-0-0
Degree 22
Conductor 32763276
Sign 0.386−0.922i0.386 - 0.922i
Analytic cond. 1.634931.63493
Root an. cond. 1.278641.27864
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  + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)25-s + (−0.5 + 0.866i)31-s + (−1.5 + 0.866i)37-s + 43-s + (−0.499 + 0.866i)49-s + (1.5 + 0.866i)67-s + (−0.5 + 0.866i)73-s + (−0.5 − 0.866i)79-s − 0.999·91-s + 2·97-s + (1.5 − 0.866i)103-s + (−1.5 − 0.866i)109-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)25-s + (−0.5 + 0.866i)31-s + (−1.5 + 0.866i)37-s + 43-s + (−0.499 + 0.866i)49-s + (1.5 + 0.866i)67-s + (−0.5 + 0.866i)73-s + (−0.5 − 0.866i)79-s − 0.999·91-s + 2·97-s + (1.5 − 0.866i)103-s + (−1.5 − 0.866i)109-s + ⋯

Functional equation

Λ(s)=(3276s/2 ΓC(s) L(s)=((0.386−0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3276s/2 ΓC(s) L(s)=((0.386−0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32763276    =    22⋅32⋅7⋅132^{2} \cdot 3^{2} \cdot 7 \cdot 13
Sign: 0.386−0.922i0.386 - 0.922i
Analytic conductor: 1.634931.63493
Root analytic conductor: 1.278641.27864
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3276(1585,⋅)\chi_{3276} (1585, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3276, ( :0), 0.386−0.922i)(2,\ 3276,\ (\ :0),\ 0.386 - 0.922i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2297191821.229719182
L(12)L(\frac12) ≈\approx 1.2297191821.229719182
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)
bad2 1 1
3 1 1
7 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
13 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good5 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
19 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
23 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
29 1+T2 1 + 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+(1.5−0.866i)T+(0.5−0.866i)T2 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}
41 1+T2 1 + T^{2}
43 1−T+T2 1 - T + T^{2}
47 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
53 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
59 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
61 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
67 1+(−1.5−0.866i)T+(0.5+0.866i)T2 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
79 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
83 1+T2 1 + T^{2}
89 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
97 1−2T+T2 1 - 2T + 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

−8.808018708222388361083739690713, −8.401355427901258069745534586826, −7.45735289571565894306087454559, −6.75956202037514628747692063659, −5.88678262867663424321559775989, −5.15096420313049167300489309310, −4.44807774789521778769943661440, −3.39270829817011179627149256740, −2.39169030920415275690909324094, −1.51273442492794027975358946607, 0.76928232423461740911054470868, 2.05766945114930847803090718769, 3.16915981514255235502836625509, 3.97272916804872554483507145888, 4.97000157823228307008810694099, 5.44319723677949011773746802858, 6.56800741298661335828446199808, 7.43222892996541527102605340630, 7.66521178533855076083032342665, 8.721428239091977409328540396038

Graph of the ZZ-function along the critical line