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Properties

Label 2-1260-105.44-c0-0-3
Degree 22
Conductor 12601260
Sign −0.286+0.958i-0.286 + 0.958i
Analytic cond. 0.6288210.628821
Root an. cond. 0.7929820.792982
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)5-s + (−0.866 − 0.5i)7-s − i·13-s + (−0.707 − 1.22i)17-s + (0.5 − 0.866i)19-s + (−0.707 + 1.22i)23-s + (0.866 − 0.499i)25-s − 1.41i·29-s + (−0.5 − 0.866i)31-s + (0.965 + 0.258i)35-s + (−0.866 − 0.5i)37-s + i·43-s + (0.707 − 1.22i)47-s + (0.499 + 0.866i)49-s + (−1.22 + 0.707i)59-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)5-s + (−0.866 − 0.5i)7-s − i·13-s + (−0.707 − 1.22i)17-s + (0.5 − 0.866i)19-s + (−0.707 + 1.22i)23-s + (0.866 − 0.499i)25-s − 1.41i·29-s + (−0.5 − 0.866i)31-s + (0.965 + 0.258i)35-s + (−0.866 − 0.5i)37-s + i·43-s + (0.707 − 1.22i)47-s + (0.499 + 0.866i)49-s + (−1.22 + 0.707i)59-s + ⋯

Functional equation

Λ(s)=(1260s/2 ΓC(s) L(s)=((−0.286+0.958i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1260s/2 ΓC(s) L(s)=((−0.286+0.958i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12601260    =    22⋅32⋅5⋅72^{2} \cdot 3^{2} \cdot 5 \cdot 7
Sign: −0.286+0.958i-0.286 + 0.958i
Analytic conductor: 0.6288210.628821
Root analytic conductor: 0.7929820.792982
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1260(989,⋅)\chi_{1260} (989, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1260, ( :0), −0.286+0.958i)(2,\ 1260,\ (\ :0),\ -0.286 + 0.958i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.54215486540.5421548654
L(12)L(\frac12) ≈\approx 0.54215486540.5421548654
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 1
3 1 1
5 1+(0.965−0.258i)T 1 + (0.965 - 0.258i)T
7 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
good11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
13 1+iT−T2 1 + iT - T^{2}
17 1+(0.707+1.22i)T+(−0.5+0.866i)T2 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2}
19 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
23 1+(0.707−1.22i)T+(−0.5−0.866i)T2 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2}
29 1+1.41iT−T2 1 + 1.41iT - T^{2}
31 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
37 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1−iT−T2 1 - iT - T^{2}
47 1+(−0.707+1.22i)T+(−0.5−0.866i)T2 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2}
53 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
59 1+(1.22−0.707i)T+(0.5−0.866i)T2 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}
61 1+(−0.5−0.866i)T2 1 + (-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+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
79 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
83 1+T2 1 + T^{2}
89 1+(−1.22−0.707i)T+(0.5+0.866i)T2 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)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

−9.652608572437304468780350723530, −8.926413044468902832014422924835, −7.68472300837040210079533583376, −7.42048225685788720632866918721, −6.45649862515445015640148528063, −5.43109437496626043576637940879, −4.32670325324356331419844706423, −3.48100328327064860352772278580, −2.62346757628067691120612645287, −0.45689532563816288200387315070, 1.79428896087623349626107626926, 3.24497150741637261208656323365, 3.99282451379923996282847357908, 4.94556141685006509697200062546, 6.13629513688679734349644112607, 6.77219149641461020901166922279, 7.71026971603166349931988538095, 8.778765446423895073357804994128, 8.950399568749888273989848067272, 10.23409557431517017649579814996

Graph of the ZZ-function along the critical line