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Properties

Label 2-3332-3332.3263-c0-0-2
Degree 22
Conductor 33323332
Sign 0.926+0.375i0.926 + 0.375i
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  + (−0.900 − 0.433i)2-s + (0.0990 + 0.433i)3-s + (0.623 + 0.781i)4-s + (0.0990 − 0.433i)6-s + (−0.222 − 0.974i)7-s + (−0.222 − 0.974i)8-s + (0.722 − 0.347i)9-s + (1.62 + 0.781i)11-s + (−0.277 + 0.347i)12-s + (1.62 + 0.781i)13-s + (−0.222 + 0.974i)14-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s − 0.801·18-s + (0.400 − 0.193i)21-s + (−1.12 − 1.40i)22-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (0.0990 + 0.433i)3-s + (0.623 + 0.781i)4-s + (0.0990 − 0.433i)6-s + (−0.222 − 0.974i)7-s + (−0.222 − 0.974i)8-s + (0.722 − 0.347i)9-s + (1.62 + 0.781i)11-s + (−0.277 + 0.347i)12-s + (1.62 + 0.781i)13-s + (−0.222 + 0.974i)14-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s − 0.801·18-s + (0.400 − 0.193i)21-s + (−1.12 − 1.40i)22-s + ⋯

Functional equation

Λ(s)=(3332s/2 ΓC(s) L(s)=((0.926+0.375i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3332s/2 ΓC(s) L(s)=((0.926+0.375i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: 0.926+0.375i0.926 + 0.375i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(3263,⋅)\chi_{3332} (3263, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3332, ( :0), 0.926+0.375i)(2,\ 3332,\ (\ :0),\ 0.926 + 0.375i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0746773171.074677317
L(12)L(\frac12) ≈\approx 1.0746773171.074677317
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+(0.900+0.433i)T 1 + (0.900 + 0.433i)T
7 1+(0.222+0.974i)T 1 + (0.222 + 0.974i)T
17 1+(−0.623+0.781i)T 1 + (-0.623 + 0.781i)T
good3 1+(−0.0990−0.433i)T+(−0.900+0.433i)T2 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2}
5 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
11 1+(−1.62−0.781i)T+(0.623+0.781i)T2 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2}
13 1+(−1.62−0.781i)T+(0.623+0.781i)T2 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2}
19 1−T2 1 - T^{2}
23 1+(1.12+1.40i)T+(−0.222+0.974i)T2 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2}
29 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
31 1+0.445T+T2 1 + 0.445T + T^{2}
37 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
41 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
43 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
47 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
53 1+(0.277+0.347i)T+(−0.222+0.974i)T2 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2}
59 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
61 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
67 1−T2 1 - T^{2}
71 1+(0.277+0.347i)T+(−0.222+0.974i)T2 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2}
73 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
79 1+1.80T+T2 1 + 1.80T + T^{2}
83 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
89 1+(−0.400+0.193i)T+(0.623−0.781i)T2 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)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

−9.064367908270269060946305347559, −8.152707662710964097548883423395, −7.26179947740138308019505865352, −6.70952275722340195837608770161, −6.16430894839717575852009848330, −4.38062174021191912654306776561, −3.99028868104225092765684075007, −3.41001227877780259508319474166, −1.82250609616824249987916364299, −1.10940514060402664446714313459, 1.27612303701564556537178517596, 1.81553218313765028260279962571, 3.25706194755196163332829296173, 4.04689507810119786006619538020, 5.71536426371494847239119233101, 5.88694964728132425346472724850, 6.56127637021715190624354710696, 7.54044513459968227206675373740, 8.259520707283768019916818097614, 8.629506462096311933827393236719

Graph of the ZZ-function along the critical line