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Properties

Label 2-1323-7.3-c0-0-0
Degree 22
Conductor 13231323
Sign 0.444+0.895i0.444 + 0.895i
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 − 1.73i·13-s + (−0.499 − 0.866i)16-s + (−0.5 + 0.866i)25-s + (1.5 + 0.866i)31-s + (−0.5 − 0.866i)37-s + 43-s + (−1.49 − 0.866i)52-s + (−1.5 + 0.866i)61-s − 0.999·64-s + (0.5 − 0.866i)67-s + (0.5 + 0.866i)79-s + 1.73i·97-s + (0.499 + 0.866i)100-s + (1.5 − 0.866i)103-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s − 1.73i·13-s + (−0.499 − 0.866i)16-s + (−0.5 + 0.866i)25-s + (1.5 + 0.866i)31-s + (−0.5 − 0.866i)37-s + 43-s + (−1.49 − 0.866i)52-s + (−1.5 + 0.866i)61-s − 0.999·64-s + (0.5 − 0.866i)67-s + (0.5 + 0.866i)79-s + 1.73i·97-s + (0.499 + 0.866i)100-s + (1.5 − 0.866i)103-s + ⋯

Functional equation

Λ(s)=(1323s/2 ΓC(s) L(s)=((0.444+0.895i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1323s/2 ΓC(s) L(s)=((0.444+0.895i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: 0.444+0.895i0.444 + 0.895i
Analytic conductor: 0.6602630.660263
Root analytic conductor: 0.8125650.812565
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1323(325,⋅)\chi_{1323} (325, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1323, ( :0), 0.444+0.895i)(2,\ 1323,\ (\ :0),\ 0.444 + 0.895i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1741483011.174148301
L(12)L(\frac12) ≈\approx 1.1741483011.174148301
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+1.73iT−T2 1 + 1.73iT - T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
19 1+(0.5−0.866i)T2 1 + (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+(−1.5−0.866i)T+(0.5+0.866i)T2 1 + (-1.5 - 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+(1.5−0.866i)T+(0.5−0.866i)T2 1 + (1.5 - 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+(0.5+0.866i)T2 1 + (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 1−1.73iT−T2 1 - 1.73iT - 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

−9.843553237222732707474108860594, −8.974816365916546114042264734303, −7.953123816083389962894659645922, −7.26190323170567542077391119980, −6.24333928291669579276411309398, −5.57645746564939591781772609965, −4.82221048394423545007869832824, −3.41928177536144544564167779469, −2.44011166590495548838277642670, −1.05989637758594156347961776512, 1.85444917602858520700677709570, 2.81924541317498148269628843430, 4.00715638032429347033310768449, 4.61935737178106725755288828837, 6.13596508737825238239272358710, 6.67264872318306544370824332564, 7.55243301252859911976984732658, 8.309353841976895473163553570025, 9.075637620829755838827725360021, 9.908404490530258951916925184214

Graph of the ZZ-function along the critical line