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Properties

Label 2-30e2-36.7-c0-0-0
Degree 22
Conductor 900900
Sign 0.939−0.342i0.939 - 0.342i
Analytic cond. 0.4491580.449158
Root an. cond. 0.6701920.670192
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  + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + 0.999·6-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.866 − 0.499i)12-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.499i)18-s − 0.999·21-s + (0.866 − 0.5i)23-s + (0.499 + 0.866i)24-s + 0.999i·27-s + 0.999i·28-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + 0.999·6-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.866 − 0.499i)12-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.499i)18-s − 0.999·21-s + (0.866 − 0.5i)23-s + (0.499 + 0.866i)24-s + 0.999i·27-s + 0.999i·28-s + ⋯

Functional equation

Λ(s)=(900s/2 ΓC(s) L(s)=((0.939−0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(900s/2 ΓC(s) L(s)=((0.939−0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 900900    =    22⋅32⋅522^{2} \cdot 3^{2} \cdot 5^{2}
Sign: 0.939−0.342i0.939 - 0.342i
Analytic conductor: 0.4491580.449158
Root analytic conductor: 0.6701920.670192
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ900(151,⋅)\chi_{900} (151, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 900, ( :0), 0.939−0.342i)(2,\ 900,\ (\ :0),\ 0.939 - 0.342i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.56777497310.5677749731
L(12)L(\frac12) ≈\approx 0.56777497310.5677749731
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+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
3 1+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
5 1 1
good7 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
13 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
17 1+T2 1 + T^{2}
19 1−T2 1 - T^{2}
23 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
29 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+T2 1 + T^{2}
41 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
43 1+(−1.73−i)T+(0.5+0.866i)T2 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2}
47 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
53 1+T2 1 + T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
67 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+T2 1 + T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
89 1−T+T2 1 - T + T^{2}
97 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)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

−10.57412507204392932604122735996, −9.489595480128461231313880652289, −8.961920343272576124571207253254, −7.974913179066763630309007449262, −7.06771007184409688590845993697, −6.07712454519151878211262297271, −5.02573130791424161470127936589, −4.08113616362854792679244335778, −2.75969226503359838135478581940, −1.29540585027610531385593859983, 1.02067260218162582006938144824, 2.22188553346566853230913988191, 4.30177137695258627828954065354, 5.34042349022179951479056514552, 5.99723983351032494088585004851, 7.19817905569662309929529293930, 7.46699946029881942192492962638, 8.449296826747297693673260201261, 9.394435600847425396542626112748, 10.42700848390621985474927384794

Graph of the ZZ-function along the critical line