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Properties

Label 2-3332-3332.407-c0-0-2
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  + (−0.222 + 0.974i)2-s + (0.777 − 0.974i)3-s + (−0.900 − 0.433i)4-s + (0.777 + 0.974i)6-s + (0.623 − 0.781i)7-s + (0.623 − 0.781i)8-s + (−0.123 − 0.541i)9-s + (0.0990 − 0.433i)11-s + (−1.12 + 0.541i)12-s + (0.0990 − 0.433i)13-s + (0.623 + 0.781i)14-s + (0.623 + 0.781i)16-s + (−0.900 + 0.433i)17-s + 0.554·18-s + (−0.277 − 1.21i)21-s + (0.400 + 0.193i)22-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (0.777 − 0.974i)3-s + (−0.900 − 0.433i)4-s + (0.777 + 0.974i)6-s + (0.623 − 0.781i)7-s + (0.623 − 0.781i)8-s + (−0.123 − 0.541i)9-s + (0.0990 − 0.433i)11-s + (−1.12 + 0.541i)12-s + (0.0990 − 0.433i)13-s + (0.623 + 0.781i)14-s + (0.623 + 0.781i)16-s + (−0.900 + 0.433i)17-s + 0.554·18-s + (−0.277 − 1.21i)21-s + (0.400 + 0.193i)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(407,⋅)\chi_{3332} (407, \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 1.4150688231.415068823
L(12)L(\frac12) ≈\approx 1.4150688231.415068823
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+(0.222−0.974i)T 1 + (0.222 - 0.974i)T
7 1+(−0.623+0.781i)T 1 + (-0.623 + 0.781i)T
17 1+(0.900−0.433i)T 1 + (0.900 - 0.433i)T
good3 1+(−0.777+0.974i)T+(−0.222−0.974i)T2 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2}
5 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
11 1+(−0.0990+0.433i)T+(−0.900−0.433i)T2 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2}
13 1+(−0.0990+0.433i)T+(−0.900−0.433i)T2 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2}
19 1−T2 1 - T^{2}
23 1+(−0.400−0.193i)T+(0.623+0.781i)T2 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2}
29 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
31 1−1.24T+T2 1 - 1.24T + T^{2}
37 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)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+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
53 1+(1.12+0.541i)T+(0.623+0.781i)T2 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2}
59 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
61 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
67 1−T2 1 - T^{2}
71 1+(1.12+0.541i)T+(0.623+0.781i)T2 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2}
73 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
79 1+0.445T+T2 1 + 0.445T + T^{2}
83 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
89 1+(0.277+1.21i)T+(−0.900+0.433i)T2 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)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

−8.396255741131433372408236078824, −8.011981819867755251076319122526, −7.35896781344060731639351647068, −6.66054513027771580912661659015, −6.06284910266007748167306223539, −4.91679078249261271172406145374, −4.25594542847401194036956686213, −3.16864521892414047524088377917, −1.92063486695949818513799242389, −0.906870331352341292476389969740, 1.56155869353784588912339738545, 2.54178433631714906095271459887, 3.16384483852969260352581428576, 4.28736852378233598901268534655, 4.59501675266173305164309397265, 5.49912617461983661980282345266, 6.74578121952291483845094408159, 7.80405078711136000542316992611, 8.520954206152058642092261634863, 9.068828447965179267013278841147

Graph of the ZZ-function along the critical line