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Properties

Label 2-3332-3332.1087-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  + (−0.733 − 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(1087,⋅)\chi_{3332} (1087, \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 0.46419423650.4641942365
L(12)L(\frac12) ≈\approx 0.46419423650.4641942365
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

−8.868130492899343896910964316653, −7.986653938622868339030153352379, −7.15152875641628122625259451384, −6.37093856353873985133058203286, −5.89348802290983894058662377324, −5.10467121986436347322614752860, −3.95951744439753687877254712444, −3.15589635607176664912106698953, −1.62410113246871487747156719430, −0.818350586665259534052658915550, 0.77137282632510650982950752861, 1.65034884004911691028303235707, 3.99000630031905019050538199583, 4.33538850906902059121965065723, 5.32717648968936041525154498722, 6.13041764339548806753547886704, 6.73875827950699445831054433318, 6.91469218898131130872438132909, 7.88530610717769833815063100538, 9.403144709646873711179553590623

Graph of the ZZ-function along the critical line