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Properties

Label 2-3332-3332.1495-c0-0-1
Degree 22
Conductor 33323332
Sign 0.801+0.598i0.801 + 0.598i
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  + (0.955 − 0.294i)2-s + (−0.266 − 0.680i)3-s + (0.826 − 0.563i)4-s + (−0.455 − 0.571i)6-s + (−0.623 + 0.781i)7-s + (0.623 − 0.781i)8-s + (0.341 − 0.317i)9-s + (1.40 + 1.29i)11-s + (−0.603 − 0.411i)12-s + (−0.425 + 1.86i)13-s + (−0.365 + 0.930i)14-s + (0.365 − 0.930i)16-s + (0.0747 − 0.997i)17-s + (0.233 − 0.403i)18-s + (0.698 + 0.215i)21-s + (1.72 + 0.829i)22-s + ⋯
L(s)  = 1  + (0.955 − 0.294i)2-s + (−0.266 − 0.680i)3-s + (0.826 − 0.563i)4-s + (−0.455 − 0.571i)6-s + (−0.623 + 0.781i)7-s + (0.623 − 0.781i)8-s + (0.341 − 0.317i)9-s + (1.40 + 1.29i)11-s + (−0.603 − 0.411i)12-s + (−0.425 + 1.86i)13-s + (−0.365 + 0.930i)14-s + (0.365 − 0.930i)16-s + (0.0747 − 0.997i)17-s + (0.233 − 0.403i)18-s + (0.698 + 0.215i)21-s + (1.72 + 0.829i)22-s + ⋯

Functional equation

Λ(s)=(3332s/2 ΓC(s) L(s)=((0.801+0.598i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3332s/2 ΓC(s) L(s)=((0.801+0.598i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: 0.801+0.598i0.801 + 0.598i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(1495,⋅)\chi_{3332} (1495, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3332, ( :0), 0.801+0.598i)(2,\ 3332,\ (\ :0),\ 0.801 + 0.598i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.2465474692.246547469
L(12)L(\frac12) ≈\approx 2.2465474692.246547469
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+(−0.955+0.294i)T 1 + (-0.955 + 0.294i)T
7 1+(0.623−0.781i)T 1 + (0.623 - 0.781i)T
17 1+(−0.0747+0.997i)T 1 + (-0.0747 + 0.997i)T
good3 1+(0.266+0.680i)T+(−0.733+0.680i)T2 1 + (0.266 + 0.680i)T + (-0.733 + 0.680i)T^{2}
5 1+(−0.955−0.294i)T2 1 + (-0.955 - 0.294i)T^{2}
11 1+(−1.40−1.29i)T+(0.0747+0.997i)T2 1 + (-1.40 - 1.29i)T + (0.0747 + 0.997i)T^{2}
13 1+(0.425−1.86i)T+(−0.900−0.433i)T2 1 + (0.425 - 1.86i)T + (-0.900 - 0.433i)T^{2}
19 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
23 1+(−0.0332−0.443i)T+(−0.988+0.149i)T2 1 + (-0.0332 - 0.443i)T + (-0.988 + 0.149i)T^{2}
29 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
31 1+(−0.623+1.07i)T+(−0.5−0.866i)T2 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2}
37 1+(−0.365−0.930i)T2 1 + (-0.365 - 0.930i)T^{2}
41 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
43 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
47 1+(−0.826+0.563i)T2 1 + (-0.826 + 0.563i)T^{2}
53 1+(1.63−1.11i)T+(0.365−0.930i)T2 1 + (1.63 - 1.11i)T + (0.365 - 0.930i)T^{2}
59 1+(−0.955+0.294i)T2 1 + (-0.955 + 0.294i)T^{2}
61 1+(−0.365−0.930i)T2 1 + (-0.365 - 0.930i)T^{2}
67 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
71 1+(1.78+0.858i)T+(0.623+0.781i)T2 1 + (1.78 + 0.858i)T + (0.623 + 0.781i)T^{2}
73 1+(−0.826−0.563i)T2 1 + (-0.826 - 0.563i)T^{2}
79 1+(0.733+1.26i)T+(−0.5+0.866i)T2 1 + (0.733 + 1.26i)T + (-0.5 + 0.866i)T^{2}
83 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
89 1+(0.535−0.496i)T+(0.0747−0.997i)T2 1 + (0.535 - 0.496i)T + (0.0747 - 0.997i)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

−9.199205246674514842318658224276, −7.46662315394007968908305174012, −6.95552046449477963964985644840, −6.53631098396123687330796528636, −5.87591839597914848311735663058, −4.62411971589807727501542588801, −4.30788202330039600581508142773, −3.15226638043441795505611865683, −2.09130813610060044664796470791, −1.44193082633587682056472683209, 1.22566605082284894282357981892, 2.97362964193008432063377579915, 3.48867612891534832571942982020, 4.23612500687160718951848652774, 5.01209024679849794518841880692, 5.85277508068631038198497304899, 6.44163803049818955634231352863, 7.17901306241707183224877304353, 8.098321046637619834752668418104, 8.669418904240607130225814318028

Graph of the ZZ-function along the critical line