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Properties

Label 2-1323-21.11-c0-0-0
Degree 22
Conductor 13231323
Sign 0.386−0.922i0.386 - 0.922i
Analytic cond. 0.6602630.660263
Root an. cond. 0.8125650.812565
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)4-s + 13-s + (−0.499 − 0.866i)16-s + (1 + 1.73i)19-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)37-s − 43-s + (−0.5 + 0.866i)52-s + (−0.5 − 0.866i)61-s + 0.999·64-s + (0.5 − 0.866i)67-s + (1 − 1.73i)73-s − 1.99·76-s + (0.5 + 0.866i)79-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + 13-s + (−0.499 − 0.866i)16-s + (1 + 1.73i)19-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)37-s − 43-s + (−0.5 + 0.866i)52-s + (−0.5 − 0.866i)61-s + 0.999·64-s + (0.5 − 0.866i)67-s + (1 − 1.73i)73-s − 1.99·76-s + (0.5 + 0.866i)79-s + ⋯

Functional equation

Λ(s)=(1323s/2 ΓC(s) L(s)=((0.386−0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1323s/2 ΓC(s) L(s)=((0.386−0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: 0.386−0.922i0.386 - 0.922i
Analytic conductor: 0.6602630.660263
Root analytic conductor: 0.8125650.812565
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1323(998,⋅)\chi_{1323} (998, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1323, ( :0), 0.386−0.922i)(2,\ 1323,\ (\ :0),\ 0.386 - 0.922i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.95827659210.9582765921
L(12)L(\frac12) ≈\approx 0.95827659210.9582765921
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
7 1 1
good2 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
5 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}
13 1−T+T2 1 - T + T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
19 1+(−1−1.73i)T+(−0.5+0.866i)T2 1 + (-1 - 1.73i)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+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 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)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)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−T2 1 - T^{2}
73 1+(−1+1.73i)T+(−0.5−0.866i)T2 1 + (-1 + 1.73i)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−T+T2 1 - T + 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

−9.799398151879063543687478145738, −9.166609446736128091737023508598, −8.203129434453055279785447843626, −7.81544293117752697062202271445, −6.77859874590492802999931802270, −5.77304106386382234872509942124, −4.87748382551209216218171225784, −3.69302898263756161925257416218, −3.27809509975445112864242636305, −1.59269155017624444001947127724, 0.918045495807824505367668021452, 2.34377151969923676383017593319, 3.71286491415362218107883439601, 4.64587814422967234179196333202, 5.50679243439660228228803765276, 6.25406186689216711603220733478, 7.14402998007331631808971790520, 8.208497512958313171949882930359, 9.035839715759652786799637475376, 9.574302464157638868787733535570

Graph of the ZZ-function along the critical line