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Properties

Label 2-150-5.4-c3-0-7
Degree 22
Conductor 150150
Sign 0.447+0.894i0.447 + 0.894i
Analytic cond. 8.850288.85028
Root an. cond. 2.974942.97494
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 + 3i·3-s − 4·4-s − 6·6-s − 32i·7-s − 8i·8-s − 9·9-s − 60·11-s − 12i·12-s − 34i·13-s + 64·14-s + 16·16-s − 42i·17-s − 18i·18-s + 76·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 1.72i·7-s − 0.353i·8-s − 0.333·9-s − 1.64·11-s − 0.288i·12-s − 0.725i·13-s + 1.22·14-s + 0.250·16-s − 0.599i·17-s − 0.235i·18-s + 0.917·19-s + ⋯

Functional equation

Λ(s)=(150s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(150s/2 ΓC(s+3/2) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 150150    =    2⋅3⋅522 \cdot 3 \cdot 5^{2}
Sign: 0.447+0.894i0.447 + 0.894i
Analytic conductor: 8.850288.85028
Root analytic conductor: 2.974942.97494
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ150(49,⋅)\chi_{150} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 150, ( :3/2), 0.447+0.894i)(2,\ 150,\ (\ :3/2),\ 0.447 + 0.894i)

Particular Values

L(2)L(2) ≈\approx 0.700717−0.433067i0.700717 - 0.433067i
L(12)L(\frac12) ≈\approx 0.700717−0.433067i0.700717 - 0.433067i
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 1−3iT 1 - 3iT
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

−12.70074530777439346284895072078, −10.91310889903058461504184134554, −10.39420403241401402880704709064, −9.341902598557670757364025634196, −7.77435807833639952018293444663, −7.36434527055723895621211508998, −5.63854425080443732257268933434, −4.63354267210361173861952079542, −3.29529185605881722530371142892, −0.38565541424593073791654779510, 1.94805511515521008373367675398, 2.98999304149075140160991469129, 5.06377722316938025968189036615, 5.97953053592003115708021121908, 7.68355861699815976034773466784, 8.670232017103985326615636378902, 9.613172678273575213380080315908, 10.93556765869403135531574427146, 11.84566126881147538660492756759, 12.62930005829372530495542724517

Graph of the ZZ-function along the critical line