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Properties

Label 2-3332-3332.1971-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  + (0.733 − 0.680i)2-s + (0.294 − 0.0444i)3-s + (0.0747 − 0.997i)4-s + (0.185 − 0.233i)6-s + (0.781 − 0.623i)7-s + (−0.623 − 0.781i)8-s + (−0.870 + 0.268i)9-s + (−1.29 − 0.400i)11-s + (−0.0222 − 0.297i)12-s + (−0.326 − 1.42i)13-s + (0.149 − 0.988i)14-s + (−0.988 − 0.149i)16-s + (0.826 − 0.563i)17-s + (−0.455 + 0.789i)18-s + (0.202 − 0.218i)21-s + (−1.22 + 0.590i)22-s + ⋯
L(s)  = 1  + (0.733 − 0.680i)2-s + (0.294 − 0.0444i)3-s + (0.0747 − 0.997i)4-s + (0.185 − 0.233i)6-s + (0.781 − 0.623i)7-s + (−0.623 − 0.781i)8-s + (−0.870 + 0.268i)9-s + (−1.29 − 0.400i)11-s + (−0.0222 − 0.297i)12-s + (−0.326 − 1.42i)13-s + (0.149 − 0.988i)14-s + (−0.988 − 0.149i)16-s + (0.826 − 0.563i)17-s + (−0.455 + 0.789i)18-s + (0.202 − 0.218i)21-s + (−1.22 + 0.590i)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(1971,⋅)\chi_{3332} (1971, \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 1.7563587291.756358729
L(12)L(\frac12) ≈\approx 1.7563587291.756358729
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+(−0.733+0.680i)T 1 + (-0.733 + 0.680i)T
7 1+(−0.781+0.623i)T 1 + (-0.781 + 0.623i)T
17 1+(−0.826+0.563i)T 1 + (-0.826 + 0.563i)T
good3 1+(−0.294+0.0444i)T+(0.955−0.294i)T2 1 + (-0.294 + 0.0444i)T + (0.955 - 0.294i)T^{2}
5 1+(0.733+0.680i)T2 1 + (0.733 + 0.680i)T^{2}
11 1+(1.29+0.400i)T+(0.826+0.563i)T2 1 + (1.29 + 0.400i)T + (0.826 + 0.563i)T^{2}
13 1+(0.326+1.42i)T+(−0.900+0.433i)T2 1 + (0.326 + 1.42i)T + (-0.900 + 0.433i)T^{2}
19 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
23 1+(−1.61−1.09i)T+(0.365+0.930i)T2 1 + (-1.61 - 1.09i)T + (0.365 + 0.930i)T^{2}
29 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
31 1+(0.781−1.35i)T+(−0.5−0.866i)T2 1 + (0.781 - 1.35i)T + (-0.5 - 0.866i)T^{2}
37 1+(0.988−0.149i)T2 1 + (0.988 - 0.149i)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+(−0.0747+0.997i)T2 1 + (-0.0747 + 0.997i)T^{2}
53 1+(−0.0546+0.728i)T+(−0.988−0.149i)T2 1 + (-0.0546 + 0.728i)T + (-0.988 - 0.149i)T^{2}
59 1+(0.733−0.680i)T2 1 + (0.733 - 0.680i)T^{2}
61 1+(0.988−0.149i)T2 1 + (0.988 - 0.149i)T^{2}
67 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
71 1+(−1.67+0.807i)T+(0.623−0.781i)T2 1 + (-1.67 + 0.807i)T + (0.623 - 0.781i)T^{2}
73 1+(−0.0747−0.997i)T2 1 + (-0.0747 - 0.997i)T^{2}
79 1+(0.294+0.510i)T+(−0.5+0.866i)T2 1 + (0.294 + 0.510i)T + (-0.5 + 0.866i)T^{2}
83 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
89 1+(−1.88+0.582i)T+(0.826−0.563i)T2 1 + (-1.88 + 0.582i)T + (0.826 - 0.563i)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

−8.414215914752464600715475529059, −7.77555323832300806656112459490, −7.17944306504826164807210930712, −5.82901349478134368267909058792, −5.18961737113302489315932100795, −4.96092224739470216469176086226, −3.40098253290197017676696325720, −3.11227707907474474955394302873, −2.09543318601966574753378176228, −0.75281708539361017126704520878, 2.09650532055160615195493078363, 2.70043378715972056883347769835, 3.75502151397471017423932353466, 4.67767455482159794741944164178, 5.32607091770622523392743490995, 5.91829963951375480858887904375, 6.86715713257095363834036002697, 7.69103101291960758305778241672, 8.144283013952964666500262578281, 8.968893554813934164594052186964

Graph of the ZZ-function along the critical line