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Properties

Label 2-3332-3332.1971-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  + (−0.733 + 0.680i)2-s + (1.95 − 0.294i)3-s + (0.0747 − 0.997i)4-s + (−1.23 + 1.54i)6-s + (0.623 + 0.781i)7-s + (0.623 + 0.781i)8-s + (2.78 − 0.858i)9-s + (−1.40 − 0.432i)11-s + (−0.147 − 1.97i)12-s + (0.326 + 1.42i)13-s + (−0.988 − 0.149i)14-s + (−0.988 − 0.149i)16-s + (0.826 − 0.563i)17-s + (−1.45 + 2.52i)18-s + (1.44 + 1.34i)21-s + (1.32 − 0.636i)22-s + ⋯
L(s)  = 1  + (−0.733 + 0.680i)2-s + (1.95 − 0.294i)3-s + (0.0747 − 0.997i)4-s + (−1.23 + 1.54i)6-s + (0.623 + 0.781i)7-s + (0.623 + 0.781i)8-s + (2.78 − 0.858i)9-s + (−1.40 − 0.432i)11-s + (−0.147 − 1.97i)12-s + (0.326 + 1.42i)13-s + (−0.988 − 0.149i)14-s + (−0.988 − 0.149i)16-s + (0.826 − 0.563i)17-s + (−1.45 + 2.52i)18-s + (1.44 + 1.34i)21-s + (1.32 − 0.636i)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(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 1.8589682941.858968294
L(12)L(\frac12) ≈\approx 1.8589682941.858968294
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+(−0.623−0.781i)T 1 + (-0.623 - 0.781i)T
17 1+(−0.826+0.563i)T 1 + (-0.826 + 0.563i)T
good3 1+(−1.95+0.294i)T+(0.955−0.294i)T2 1 + (-1.95 + 0.294i)T + (0.955 - 0.294i)T^{2}
5 1+(0.733+0.680i)T2 1 + (0.733 + 0.680i)T^{2}
11 1+(1.40+0.432i)T+(0.826+0.563i)T2 1 + (1.40 + 0.432i)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+(0.367+0.250i)T+(0.365+0.930i)T2 1 + (0.367 + 0.250i)T + (0.365 + 0.930i)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+(0.988−0.149i)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+(0.658−0.317i)T+(0.623−0.781i)T2 1 + (0.658 - 0.317i)T + (0.623 - 0.781i)T^{2}
73 1+(−0.0747−0.997i)T2 1 + (-0.0747 - 0.997i)T^{2}
79 1+(0.955+1.65i)T+(−0.5+0.866i)T2 1 + (0.955 + 1.65i)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

−8.568906142170053324462363705657, −8.352184802807978173043066117418, −7.60851346583044570362264333151, −7.08035465911243357182362982162, −6.10067651157915969865619708403, −5.10399599114579954828302882590, −4.23962366134357909752512867211, −3.02421612615896916189864337216, −2.22545697546608737360943963645, −1.54172678863558045964134210776, 1.35612012906003454602534806278, 2.23748452126035914842930989058, 3.05841681744816473345643522186, 3.70606342822045487934250675000, 4.41558787284496910939108645033, 5.51001513911989508036411613555, 7.32497587525810460632487716128, 7.58955916229801554869883680725, 8.105517268152289020637854959102, 8.536747489391904120441557406183

Graph of the ZZ-function along the critical line