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Properties

Label 2-675-27.4-c1-0-30
Degree 22
Conductor 675675
Sign −0.835+0.549i-0.835 + 0.549i
Analytic cond. 5.389905.38990
Root an. cond. 2.321612.32161
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.826 − 0.300i)2-s + (0.592 + 1.62i)3-s + (−0.939 − 0.788i)4-s − 1.52i·6-s + (−2.87 + 2.41i)7-s + (1.41 + 2.45i)8-s + (−2.29 + 1.92i)9-s + (−0.180 − 1.02i)11-s + (0.726 − 1.99i)12-s + (2.99 − 1.08i)13-s + (3.10 − 1.13i)14-s + (−0.00727 − 0.0412i)16-s + (−0.233 + 0.405i)17-s + (2.47 − 0.902i)18-s + (−2.34 − 4.06i)19-s + ⋯
L(s)  = 1  + (−0.584 − 0.212i)2-s + (0.342 + 0.939i)3-s + (−0.469 − 0.394i)4-s − 0.621i·6-s + (−1.08 + 0.913i)7-s + (0.501 + 0.868i)8-s + (−0.766 + 0.642i)9-s + (−0.0545 − 0.309i)11-s + (0.209 − 0.576i)12-s + (0.830 − 0.302i)13-s + (0.830 − 0.302i)14-s + (−0.00181 − 0.0103i)16-s + (−0.0567 + 0.0982i)17-s + (0.584 − 0.212i)18-s + (−0.538 − 0.932i)19-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=((−0.835+0.549i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+1/2) L(s)=((−0.835+0.549i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: −0.835+0.549i-0.835 + 0.549i
Analytic conductor: 5.389905.38990
Root analytic conductor: 2.321612.32161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ675(301,⋅)\chi_{675} (301, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 675, ( :1/2), −0.835+0.549i)(2,\ 675,\ (\ :1/2),\ -0.835 + 0.549i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) 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)
bad3 1+(−0.592−1.62i)T 1 + (-0.592 - 1.62i)T
5 1 1
good2 1+(0.826+0.300i)T+(1.53+1.28i)T2 1 + (0.826 + 0.300i)T + (1.53 + 1.28i)T^{2}
7 1+(2.87−2.41i)T+(1.21−6.89i)T2 1 + (2.87 - 2.41i)T + (1.21 - 6.89i)T^{2}
11 1+(0.180+1.02i)T+(−10.3+3.76i)T2 1 + (0.180 + 1.02i)T + (-10.3 + 3.76i)T^{2}
13 1+(−2.99+1.08i)T+(9.95−8.35i)T2 1 + (-2.99 + 1.08i)T + (9.95 - 8.35i)T^{2}
17 1+(0.233−0.405i)T+(−8.5−14.7i)T2 1 + (0.233 - 0.405i)T + (-8.5 - 14.7i)T^{2}
19 1+(2.34+4.06i)T+(−9.5+16.4i)T2 1 + (2.34 + 4.06i)T + (-9.5 + 16.4i)T^{2}
23 1+(4.11+3.45i)T+(3.99+22.6i)T2 1 + (4.11 + 3.45i)T + (3.99 + 22.6i)T^{2}
29 1+(5.45+1.98i)T+(22.2+18.6i)T2 1 + (5.45 + 1.98i)T + (22.2 + 18.6i)T^{2}
31 1+(3.14+2.63i)T+(5.38+30.5i)T2 1 + (3.14 + 2.63i)T + (5.38 + 30.5i)T^{2}
37 1+(−2.23+3.87i)T+(−18.5−32.0i)T2 1 + (-2.23 + 3.87i)T + (-18.5 - 32.0i)T^{2}
41 1+(7.52−2.73i)T+(31.4−26.3i)T2 1 + (7.52 - 2.73i)T + (31.4 - 26.3i)T^{2}
43 1+(−2.11−11.9i)T+(−40.4+14.7i)T2 1 + (-2.11 - 11.9i)T + (-40.4 + 14.7i)T^{2}
47 1+(2.65−2.22i)T+(8.16−46.2i)T2 1 + (2.65 - 2.22i)T + (8.16 - 46.2i)T^{2}
53 1+8.83T+53T2 1 + 8.83T + 53T^{2}
59 1+(−2.36+13.4i)T+(−55.4−20.1i)T2 1 + (-2.36 + 13.4i)T + (-55.4 - 20.1i)T^{2}
61 1+(−7.46+6.26i)T+(10.5−60.0i)T2 1 + (-7.46 + 6.26i)T + (10.5 - 60.0i)T^{2}
67 1+(1.71−0.623i)T+(51.3−43.0i)T2 1 + (1.71 - 0.623i)T + (51.3 - 43.0i)T^{2}
71 1+(3.85−6.67i)T+(−35.5−61.4i)T2 1 + (3.85 - 6.67i)T + (-35.5 - 61.4i)T^{2}
73 1+(−0.407−0.705i)T+(−36.5+63.2i)T2 1 + (-0.407 - 0.705i)T + (-36.5 + 63.2i)T^{2}
79 1+(−3.81−1.38i)T+(60.5+50.7i)T2 1 + (-3.81 - 1.38i)T + (60.5 + 50.7i)T^{2}
83 1+(15.9+5.81i)T+(63.5+53.3i)T2 1 + (15.9 + 5.81i)T + (63.5 + 53.3i)T^{2}
89 1+(−5.19−9.00i)T+(−44.5+77.0i)T2 1 + (-5.19 - 9.00i)T + (-44.5 + 77.0i)T^{2}
97 1+(−1.06−6.02i)T+(−91.1+33.1i)T2 1 + (-1.06 - 6.02i)T + (-91.1 + 33.1i)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

−9.771254242509671863606291340351, −9.508230002400118159576195474376, −8.630962218181661175103712127717, −8.086805866626885886496061631208, −6.30494894614235658450182959236, −5.63749118667051617947863283401, −4.53955848223434342618902710574, −3.41501094099480271649277250043, −2.26685790394321552586355868351, 0, 1.57965517537069703265762732360, 3.41549439983230138481795264119, 3.97683593028085985336245776085, 5.78946081438974505556396900205, 6.83246891799800147289380519979, 7.32298780234918389023675800660, 8.244974253932667295273491889207, 8.979493290676816420947654172167, 9.815242838639603366258973141456

Graph of the ZZ-function along the critical line