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Properties

Label 2-3332-3332.1563-c0-0-1
Degree 22
Conductor 33323332
Sign 0.999+0.0213i0.999 + 0.0213i
Analytic cond. 1.662881.66288
Root an. cond. 1.289521.28952
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  + (0.988 − 0.149i)2-s + (−0.930 − 0.634i)3-s + (0.955 − 0.294i)4-s + (−1.01 − 0.488i)6-s + (0.433 + 0.900i)7-s + (0.900 − 0.433i)8-s + (0.0983 + 0.250i)9-s + (−0.108 + 0.277i)11-s + (−1.07 − 0.332i)12-s + (1.23 + 1.54i)13-s + (0.563 + 0.826i)14-s + (0.826 − 0.563i)16-s + (−0.733 + 0.680i)17-s + (0.134 + 0.233i)18-s + (0.167 − 1.11i)21-s + (−0.0663 + 0.290i)22-s + ⋯
L(s)  = 1  + (0.988 − 0.149i)2-s + (−0.930 − 0.634i)3-s + (0.955 − 0.294i)4-s + (−1.01 − 0.488i)6-s + (0.433 + 0.900i)7-s + (0.900 − 0.433i)8-s + (0.0983 + 0.250i)9-s + (−0.108 + 0.277i)11-s + (−1.07 − 0.332i)12-s + (1.23 + 1.54i)13-s + (0.563 + 0.826i)14-s + (0.826 − 0.563i)16-s + (−0.733 + 0.680i)17-s + (0.134 + 0.233i)18-s + (0.167 − 1.11i)21-s + (−0.0663 + 0.290i)22-s + ⋯

Functional equation

Λ(s)=(3332s/2 ΓC(s) L(s)=((0.999+0.0213i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3332s/2 ΓC(s) L(s)=((0.999+0.0213i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: 0.999+0.0213i0.999 + 0.0213i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(1563,⋅)\chi_{3332} (1563, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3332, ( :0), 0.999+0.0213i)(2,\ 3332,\ (\ :0),\ 0.999 + 0.0213i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.9052165201.905216520
L(12)L(\frac12) ≈\approx 1.9052165201.905216520
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+(−0.988+0.149i)T 1 + (-0.988 + 0.149i)T
7 1+(−0.433−0.900i)T 1 + (-0.433 - 0.900i)T
17 1+(0.733−0.680i)T 1 + (0.733 - 0.680i)T
good3 1+(0.930+0.634i)T+(0.365+0.930i)T2 1 + (0.930 + 0.634i)T + (0.365 + 0.930i)T^{2}
5 1+(0.988+0.149i)T2 1 + (0.988 + 0.149i)T^{2}
11 1+(0.108−0.277i)T+(−0.733−0.680i)T2 1 + (0.108 - 0.277i)T + (-0.733 - 0.680i)T^{2}
13 1+(−1.23−1.54i)T+(−0.222+0.974i)T2 1 + (-1.23 - 1.54i)T + (-0.222 + 0.974i)T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1+(−1.14−1.06i)T+(0.0747+0.997i)T2 1 + (-1.14 - 1.06i)T + (0.0747 + 0.997i)T^{2}
29 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
31 1+(0.433+0.751i)T+(−0.5+0.866i)T2 1 + (0.433 + 0.751i)T + (-0.5 + 0.866i)T^{2}
37 1+(−0.826−0.563i)T2 1 + (-0.826 - 0.563i)T^{2}
41 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
43 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
47 1+(−0.955+0.294i)T2 1 + (-0.955 + 0.294i)T^{2}
53 1+(−0.142+0.0440i)T+(0.826−0.563i)T2 1 + (-0.142 + 0.0440i)T + (0.826 - 0.563i)T^{2}
59 1+(0.988−0.149i)T2 1 + (0.988 - 0.149i)T^{2}
61 1+(−0.826−0.563i)T2 1 + (-0.826 - 0.563i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1+(−0.443+1.94i)T+(−0.900−0.433i)T2 1 + (-0.443 + 1.94i)T + (-0.900 - 0.433i)T^{2}
73 1+(−0.955−0.294i)T2 1 + (-0.955 - 0.294i)T^{2}
79 1+(−0.930+1.61i)T+(−0.5−0.866i)T2 1 + (-0.930 + 1.61i)T + (-0.5 - 0.866i)T^{2}
83 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
89 1+(0.603+1.53i)T+(−0.733+0.680i)T2 1 + (0.603 + 1.53i)T + (-0.733 + 0.680i)T^{2}
97 1−T2 1 - 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

−8.869588203443944849972650333485, −7.78917275938107207739490919930, −6.99323820144177553010275988145, −6.25880288436686877505895326910, −5.92850403891973733299358375918, −5.09732227874072791511161481757, −4.26945921441735921497853360271, −3.40992449925509977358027963309, −2.02493945710192283222924393453, −1.53432394649444765626658820204, 1.01103626063230301384191423904, 2.58933042287587042916739728316, 3.59343409956983852682922160116, 4.26752283547400761890879182960, 5.13040087743797182642496593630, 5.49396509038003610043160445917, 6.38842605655296625782685551100, 7.06414835668596702404265324653, 7.987088572267359132766412295056, 8.544854669448180681620023059006

Graph of the ZZ-function along the critical line