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Properties

Label 2-3332-3332.1563-c0-0-2
Degree 22
Conductor 33323332
Sign −0.999−0.0213i-0.999 - 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 + (−1.36 − 0.930i)3-s + (0.955 − 0.294i)4-s + (1.48 + 0.716i)6-s + (0.900 − 0.433i)7-s + (−0.900 + 0.433i)8-s + (0.632 + 1.61i)9-s + (0.722 − 1.84i)11-s + (−1.57 − 0.487i)12-s + (−1.23 − 1.54i)13-s + (−0.826 + 0.563i)14-s + (0.826 − 0.563i)16-s + (−0.733 + 0.680i)17-s + (−0.865 − 1.49i)18-s + (−1.63 − 0.246i)21-s + (−0.440 + 1.92i)22-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)2-s + (−1.36 − 0.930i)3-s + (0.955 − 0.294i)4-s + (1.48 + 0.716i)6-s + (0.900 − 0.433i)7-s + (−0.900 + 0.433i)8-s + (0.632 + 1.61i)9-s + (0.722 − 1.84i)11-s + (−1.57 − 0.487i)12-s + (−1.23 − 1.54i)13-s + (−0.826 + 0.563i)14-s + (0.826 − 0.563i)16-s + (−0.733 + 0.680i)17-s + (−0.865 − 1.49i)18-s + (−1.63 − 0.246i)21-s + (−0.440 + 1.92i)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: −0.999−0.0213i-0.999 - 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 0.35379615820.3537961582
L(12)L(\frac12) ≈\approx 0.35379615820.3537961582
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+(−0.900+0.433i)T 1 + (-0.900 + 0.433i)T
17 1+(0.733−0.680i)T 1 + (0.733 - 0.680i)T
good3 1+(1.36+0.930i)T+(0.365+0.930i)T2 1 + (1.36 + 0.930i)T + (0.365 + 0.930i)T^{2}
5 1+(0.988+0.149i)T2 1 + (0.988 + 0.149i)T^{2}
11 1+(−0.722+1.84i)T+(−0.733−0.680i)T2 1 + (-0.722 + 1.84i)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+(−0.914−0.848i)T+(0.0747+0.997i)T2 1 + (-0.914 - 0.848i)T + (0.0747 + 0.997i)T^{2}
29 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
31 1+(0.900+1.56i)T+(−0.5+0.866i)T2 1 + (0.900 + 1.56i)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+(−0.0332+0.145i)T+(−0.900−0.433i)T2 1 + (-0.0332 + 0.145i)T + (-0.900 - 0.433i)T^{2}
73 1+(−0.955−0.294i)T2 1 + (-0.955 - 0.294i)T^{2}
79 1+(−0.365+0.632i)T+(−0.5−0.866i)T2 1 + (-0.365 + 0.632i)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

−8.144142521591522882705506410555, −7.69902851501753271415373516423, −7.09268378297203453487922965574, −6.14522106836857224710555371569, −5.72792327033924935058227998708, −5.06931848759168048901161940803, −3.60963968349958761813752442029, −2.27964117513428697604530071209, −1.22187240350877193597530093889, −0.38194429820159092299038854588, 1.60605356429206217793828883695, 2.36006277633540687133552307198, 4.05535497323619597215368257682, 4.71624792654173289738878429580, 5.17451735656388856794913972940, 6.43447165772486113743304394444, 6.96437364361491903784975747126, 7.48115320577441897959843129125, 8.913132234252971101104060558532, 9.236339687777072303029484121999

Graph of the ZZ-function along the critical line