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Properties

Label 2-1350-45.4-c1-0-0
Degree 22
Conductor 13501350
Sign −0.980−0.195i-0.980 - 0.195i
Analytic cond. 10.779810.7798
Root an. cond. 3.283263.28326
Motivic weight 11
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 + (0.499 − 0.866i)4-s + (0.389 − 0.224i)7-s + 0.999i·8-s + (−2.44 − 4.24i)11-s + (0.389 + 0.224i)13-s + (−0.224 + 0.389i)14-s + (−0.5 − 0.866i)16-s + 4.89i·17-s − 7.44·19-s + (4.24 + 2.44i)22-s + (−2.12 − 1.22i)23-s − 0.449·26-s − 0.449i·28-s + (1.22 + 2.12i)29-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.147 − 0.0849i)7-s + 0.353i·8-s + (−0.738 − 1.27i)11-s + (0.107 + 0.0623i)13-s + (−0.0600 + 0.104i)14-s + (−0.125 − 0.216i)16-s + 1.18i·17-s − 1.70·19-s + (0.904 + 0.522i)22-s + (−0.442 − 0.255i)23-s − 0.0881·26-s − 0.0849i·28-s + (0.227 + 0.393i)29-s + ⋯

Functional equation

Λ(s)=(1350s/2 ΓC(s) L(s)=((−0.980−0.195i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1350s/2 ΓC(s+1/2) L(s)=((−0.980−0.195i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13501350    =    2⋅33⋅522 \cdot 3^{3} \cdot 5^{2}
Sign: −0.980−0.195i-0.980 - 0.195i
Analytic conductor: 10.779810.7798
Root analytic conductor: 3.283263.28326
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1350(199,⋅)\chi_{1350} (199, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1350, ( :1/2), −0.980−0.195i)(2,\ 1350,\ (\ :1/2),\ -0.980 - 0.195i)

Particular Values

L(1)L(1) ≈\approx 0.21306834170.2130683417
L(12)L(\frac12) ≈\approx 0.21306834170.2130683417
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)
bad2 1+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
3 1 1
5 1 1
good7 1+(−0.389+0.224i)T+(3.5−6.06i)T2 1 + (-0.389 + 0.224i)T + (3.5 - 6.06i)T^{2}
11 1+(2.44+4.24i)T+(−5.5+9.52i)T2 1 + (2.44 + 4.24i)T + (-5.5 + 9.52i)T^{2}
13 1+(−0.389−0.224i)T+(6.5+11.2i)T2 1 + (-0.389 - 0.224i)T + (6.5 + 11.2i)T^{2}
17 1−4.89iT−17T2 1 - 4.89iT - 17T^{2}
19 1+7.44T+19T2 1 + 7.44T + 19T^{2}
23 1+(2.12+1.22i)T+(11.5+19.9i)T2 1 + (2.12 + 1.22i)T + (11.5 + 19.9i)T^{2}
29 1+(−1.22−2.12i)T+(−14.5+25.1i)T2 1 + (-1.22 - 2.12i)T + (-14.5 + 25.1i)T^{2}
31 1+(2.22−3.85i)T+(−15.5−26.8i)T2 1 + (2.22 - 3.85i)T + (-15.5 - 26.8i)T^{2}
37 1−11.3iT−37T2 1 - 11.3iT - 37T^{2}
41 1+(−4.5+7.79i)T+(−20.5−35.5i)T2 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2}
43 1+(−2.20+1.27i)T+(21.5−37.2i)T2 1 + (-2.20 + 1.27i)T + (21.5 - 37.2i)T^{2}
47 1+(9.43−5.44i)T+(23.5−40.7i)T2 1 + (9.43 - 5.44i)T + (23.5 - 40.7i)T^{2}
53 1−3.55iT−53T2 1 - 3.55iT - 53T^{2}
59 1+(−2.72+4.71i)T+(−29.5−51.0i)T2 1 + (-2.72 + 4.71i)T + (-29.5 - 51.0i)T^{2}
61 1+(4+6.92i)T+(−30.5+52.8i)T2 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2}
67 1+(−0.301−0.174i)T+(33.5+58.0i)T2 1 + (-0.301 - 0.174i)T + (33.5 + 58.0i)T^{2}
71 1+13.3T+71T2 1 + 13.3T + 71T^{2}
73 1+iT−73T2 1 + iT - 73T^{2}
79 1+(−8.34−14.4i)T+(−39.5+68.4i)T2 1 + (-8.34 - 14.4i)T + (-39.5 + 68.4i)T^{2}
83 1+(−4.71+2.72i)T+(41.5−71.8i)T2 1 + (-4.71 + 2.72i)T + (41.5 - 71.8i)T^{2}
89 1+9T+89T2 1 + 9T + 89T^{2}
97 1+(7.61−4.39i)T+(48.5−84.0i)T2 1 + (7.61 - 4.39i)T + (48.5 - 84.0i)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

−10.04182254225589312621368954788, −8.917825423181372554618526657774, −8.339490578109850870881137555081, −7.86835728255680159585753695233, −6.56416748692617554562181564673, −6.12498154022965176259003090909, −5.11141767083487073126422444153, −3.99685387688611163999115370024, −2.80411736353517922731003428963, −1.51733003721263135997844622854, 0.10246353154306206935961105424, 1.91121521712688236787892436783, 2.64111953416455343705167776431, 4.05124396753137062726543635823, 4.85561133051351852743234338798, 5.99030135531057856854626751787, 7.03425187799792393197020344952, 7.67200150899748574084827668781, 8.453505232763313032449521991846, 9.373442301791414041863858157950

Graph of the ZZ-function along the critical line