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Properties

Label 2-3332-476.263-c0-0-1
Degree 22
Conductor 33323332
Sign 0.123−0.992i0.123 - 0.992i
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  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (0.0999 + 0.758i)5-s + (0.707 + 0.707i)8-s + (−0.258 + 0.965i)9-s + (−0.0999 + 0.758i)10-s + (0.500 + 0.866i)16-s + (−0.258 − 0.965i)17-s + (−0.499 + 0.866i)18-s + (−0.292 + 0.707i)20-s + (0.400 − 0.107i)25-s + (0.707 + 0.292i)29-s + (0.258 + 0.965i)32-s − i·34-s + (−0.707 + 0.707i)36-s + (−0.758 + 0.0999i)37-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (0.0999 + 0.758i)5-s + (0.707 + 0.707i)8-s + (−0.258 + 0.965i)9-s + (−0.0999 + 0.758i)10-s + (0.500 + 0.866i)16-s + (−0.258 − 0.965i)17-s + (−0.499 + 0.866i)18-s + (−0.292 + 0.707i)20-s + (0.400 − 0.107i)25-s + (0.707 + 0.292i)29-s + (0.258 + 0.965i)32-s − i·34-s + (−0.707 + 0.707i)36-s + (−0.758 + 0.0999i)37-s + ⋯

Functional equation

Λ(s)=(3332s/2 ΓC(s) L(s)=((0.123−0.992i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3332s/2 ΓC(s) L(s)=((0.123−0.992i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: 0.123−0.992i0.123 - 0.992i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(263,⋅)\chi_{3332} (263, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3332, ( :0), 0.123−0.992i)(2,\ 3332,\ (\ :0),\ 0.123 - 0.992i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.3402398832.340239883
L(12)L(\frac12) ≈\approx 2.3402398832.340239883
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+(−0.965−0.258i)T 1 + (-0.965 - 0.258i)T
7 1 1
17 1+(0.258+0.965i)T 1 + (0.258 + 0.965i)T
good3 1+(0.258−0.965i)T2 1 + (0.258 - 0.965i)T^{2}
5 1+(−0.0999−0.758i)T+(−0.965+0.258i)T2 1 + (-0.0999 - 0.758i)T + (-0.965 + 0.258i)T^{2}
11 1+(−0.965−0.258i)T2 1 + (-0.965 - 0.258i)T^{2}
13 1−T2 1 - T^{2}
19 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
23 1+(0.258+0.965i)T2 1 + (0.258 + 0.965i)T^{2}
29 1+(−0.707−0.292i)T+(0.707+0.707i)T2 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2}
31 1+(0.258−0.965i)T2 1 + (0.258 - 0.965i)T^{2}
37 1+(0.758−0.0999i)T+(0.965−0.258i)T2 1 + (0.758 - 0.0999i)T + (0.965 - 0.258i)T^{2}
41 1+(1.70−0.707i)T+(0.707−0.707i)T2 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2}
43 1−iT2 1 - iT^{2}
47 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
53 1+(0.366+1.36i)T+(−0.866+0.5i)T2 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2}
59 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
61 1+(−1.46−1.12i)T+(0.258+0.965i)T2 1 + (-1.46 - 1.12i)T + (0.258 + 0.965i)T^{2}
67 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
71 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
73 1+(0.607−0.465i)T+(0.258−0.965i)T2 1 + (0.607 - 0.465i)T + (0.258 - 0.965i)T^{2}
79 1+(0.258+0.965i)T2 1 + (0.258 + 0.965i)T^{2}
83 1+iT2 1 + iT^{2}
89 1+(−1.73+i)T+(0.5−0.866i)T2 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2}
97 1+(0.707+0.292i)T+(0.707+0.707i)T2 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)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

−8.664523605165466202877320599531, −8.147375211143371882560108155336, −7.05828617196057181011390131756, −6.92324425209449342205079439440, −5.89533588073568944849463367848, −5.09451813857499216911187577111, −4.56424734493325392479743343890, −3.34899124692051940482125257461, −2.76597752983309755168459772768, −1.86137447689934077097655311697, 1.05740136694526897986230506952, 2.11276847089133480192130302580, 3.28046304674850454497978475998, 3.91812786811348459960255313787, 4.81926753746392091228970635769, 5.46044152287244802555162706659, 6.34171621573562970195706942602, 6.78410677624725477064403623972, 7.88705168490309106426204937911, 8.718121995825093886757082408793

Graph of the ZZ-function along the critical line