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Properties

Label 2-450-5.4-c3-0-18
Degree 22
Conductor 450450
Sign −0.894+0.447i-0.894 + 0.447i
Analytic cond. 26.550826.5508
Root an. cond. 5.152755.15275
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s − 4·4-s − 32i·7-s + 8i·8-s + 60·11-s − 34i·13-s − 64·14-s + 16·16-s + 42i·17-s + 76·19-s − 120i·22-s − 68·26-s + 128i·28-s + 6·29-s − 232·31-s − 32i·32-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 1.72i·7-s + 0.353i·8-s + 1.64·11-s − 0.725i·13-s − 1.22·14-s + 0.250·16-s + 0.599i·17-s + 0.917·19-s − 1.16i·22-s − 0.512·26-s + 0.863i·28-s + 0.0384·29-s − 1.34·31-s − 0.176i·32-s + ⋯

Functional equation

Λ(s)=(450s/2 ΓC(s) L(s)=((−0.894+0.447i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(450s/2 ΓC(s+3/2) L(s)=((−0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 450450    =    2⋅32⋅522 \cdot 3^{2} \cdot 5^{2}
Sign: −0.894+0.447i-0.894 + 0.447i
Analytic conductor: 26.550826.5508
Root analytic conductor: 5.152755.15275
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ450(199,⋅)\chi_{450} (199, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 450, ( :3/2), −0.894+0.447i)(2,\ 450,\ (\ :3/2),\ -0.894 + 0.447i)

Particular Values

L(2)L(2) ≈\approx 1.7004990781.700499078
L(12)L(\frac12) ≈\approx 1.7004990781.700499078
L(52)L(\frac{5}{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+2iT 1 + 2iT
3 1 1
5 1 1
good7 1+32iT−343T2 1 + 32iT - 343T^{2}
11 1−60T+1.33e3T2 1 - 60T + 1.33e3T^{2}
13 1+34iT−2.19e3T2 1 + 34iT - 2.19e3T^{2}
17 1−42iT−4.91e3T2 1 - 42iT - 4.91e3T^{2}
19 1−76T+6.85e3T2 1 - 76T + 6.85e3T^{2}
23 1−1.21e4T2 1 - 1.21e4T^{2}
29 1−6T+2.43e4T2 1 - 6T + 2.43e4T^{2}
31 1+232T+2.97e4T2 1 + 232T + 2.97e4T^{2}
37 1+134iT−5.06e4T2 1 + 134iT - 5.06e4T^{2}
41 1+234T+6.89e4T2 1 + 234T + 6.89e4T^{2}
43 1+412iT−7.95e4T2 1 + 412iT - 7.95e4T^{2}
47 1+360iT−1.03e5T2 1 + 360iT - 1.03e5T^{2}
53 1+222iT−1.48e5T2 1 + 222iT - 1.48e5T^{2}
59 1−660T+2.05e5T2 1 - 660T + 2.05e5T^{2}
61 1+490T+2.26e5T2 1 + 490T + 2.26e5T^{2}
67 1+812iT−3.00e5T2 1 + 812iT - 3.00e5T^{2}
71 1+120T+3.57e5T2 1 + 120T + 3.57e5T^{2}
73 1−746iT−3.89e5T2 1 - 746iT - 3.89e5T^{2}
79 1+152T+4.93e5T2 1 + 152T + 4.93e5T^{2}
83 1−804iT−5.71e5T2 1 - 804iT - 5.71e5T^{2}
89 1+678T+7.04e5T2 1 + 678T + 7.04e5T^{2}
97 1+194iT−9.12e5T2 1 + 194iT - 9.12e5T^{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

−10.36960486207413218562429654310, −9.635198483667549174820465745977, −8.629838067292762559393991131701, −7.48408825553197541344651142689, −6.71176049185541299580077309044, −5.31839359331918009536537271833, −3.94683224588907942385580563854, −3.57013973419933974095619786190, −1.61124761801604400487030980137, −0.59461831291699184534200897269, 1.56047244359457693387529961113, 3.10864917349973933654939251243, 4.47828883224516753490115609939, 5.56132103344646327153951873948, 6.34031533826960366872582671706, 7.24285252541707020506546643841, 8.504650111796543161488609042482, 9.200758631150008014799649441554, 9.628846829947841016113485089352, 11.41634391908890801855662568750

Graph of the ZZ-function along the critical line