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Properties

Label 2-3332-3332.1359-c0-0-3
Degree 22
Conductor 33323332
Sign −0.801+0.598i-0.801 + 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 + (−0.974 − 1.22i)3-s + (−0.900 + 0.433i)4-s + (0.974 − 1.22i)6-s + (0.781 − 0.623i)7-s + (−0.623 − 0.781i)8-s + (−0.321 + 1.40i)9-s + (−0.433 − 1.90i)11-s + (1.40 + 0.678i)12-s + (−0.0990 − 0.433i)13-s + (0.781 + 0.623i)14-s + (0.623 − 0.781i)16-s + (−0.900 − 0.433i)17-s − 1.44·18-s + (−1.52 − 0.347i)21-s + (1.75 − 0.846i)22-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)2-s + (−0.974 − 1.22i)3-s + (−0.900 + 0.433i)4-s + (0.974 − 1.22i)6-s + (0.781 − 0.623i)7-s + (−0.623 − 0.781i)8-s + (−0.321 + 1.40i)9-s + (−0.433 − 1.90i)11-s + (1.40 + 0.678i)12-s + (−0.0990 − 0.433i)13-s + (0.781 + 0.623i)14-s + (0.623 − 0.781i)16-s + (−0.900 − 0.433i)17-s − 1.44·18-s + (−1.52 − 0.347i)21-s + (1.75 − 0.846i)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+0.598i-0.801 + 0.598i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(1359,⋅)\chi_{3332} (1359, \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 0.42735833820.4273583382
L(12)L(\frac12) ≈\approx 0.42735833820.4273583382
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+(−0.781+0.623i)T 1 + (-0.781 + 0.623i)T
17 1+(0.900+0.433i)T 1 + (0.900 + 0.433i)T
good3 1+(0.974+1.22i)T+(−0.222+0.974i)T2 1 + (0.974 + 1.22i)T + (-0.222 + 0.974i)T^{2}
5 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
11 1+(0.433+1.90i)T+(−0.900+0.433i)T2 1 + (0.433 + 1.90i)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+(1.75−0.846i)T+(0.623−0.781i)T2 1 + (1.75 - 0.846i)T + (0.623 - 0.781i)T^{2}
29 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
31 1−1.56T+T2 1 - 1.56T + 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+(1.40−0.678i)T+(0.623−0.781i)T2 1 + (1.40 - 0.678i)T + (0.623 - 0.781i)T^{2}
73 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
79 1+1.94T+T2 1 + 1.94T + 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

−8.104545540949305425919616301160, −7.70504125710573675664340503245, −7.01288259300451560212850498626, −6.11543972583117048913994141812, −5.79793878195699069812163193371, −5.03638973199086095671098649627, −4.08303706359951106917795204764, −2.96096198791885684176359945591, −1.40432744376348382392468374215, −0.27448086789833278714479463189, 1.86710255932552172016212034657, 2.50349385747068693840208406740, 4.06081075910682148603794353477, 4.60548817932567424209086391866, 4.76474603490040270418594549273, 5.84886449894446984705956300999, 6.47398277543495059303142276961, 7.87661464792931116646939910901, 8.607750376940455559255080502022, 9.460091179807245214618354741875

Graph of the ZZ-function along the critical line