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Properties

Label 2-1512-21.5-c1-0-9
Degree 22
Conductor 15121512
Sign 0.562−0.826i0.562 - 0.826i
Analytic cond. 12.073312.0733
Root an. cond. 3.474673.47467
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  + (−0.310 + 0.537i)5-s + (−1.44 − 2.21i)7-s + (−1.91 + 1.10i)11-s + 0.963i·13-s + (0.653 + 1.13i)17-s + (−1.57 − 0.911i)19-s + (5.27 + 3.04i)23-s + (2.30 + 3.99i)25-s + 6.68i·29-s + (3.48 − 2.01i)31-s + (1.63 − 0.0859i)35-s + (0.165 − 0.286i)37-s + 11.7·41-s − 2.18·43-s + (−1.27 + 2.20i)47-s + ⋯
L(s)  = 1  + (−0.138 + 0.240i)5-s + (−0.544 − 0.838i)7-s + (−0.577 + 0.333i)11-s + 0.267i·13-s + (0.158 + 0.274i)17-s + (−0.362 − 0.209i)19-s + (1.10 + 0.635i)23-s + (0.461 + 0.799i)25-s + 1.24i·29-s + (0.626 − 0.361i)31-s + (0.277 − 0.0145i)35-s + (0.0271 − 0.0470i)37-s + 1.83·41-s − 0.333·43-s + (−0.185 + 0.321i)47-s + ⋯

Functional equation

Λ(s)=(1512s/2 ΓC(s) L(s)=((0.562−0.826i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 - 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1512s/2 ΓC(s+1/2) L(s)=((0.562−0.826i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15121512    =    23⋅33⋅72^{3} \cdot 3^{3} \cdot 7
Sign: 0.562−0.826i0.562 - 0.826i
Analytic conductor: 12.073312.0733
Root analytic conductor: 3.474673.47467
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1512(593,⋅)\chi_{1512} (593, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1512, ( :1/2), 0.562−0.826i)(2,\ 1512,\ (\ :1/2),\ 0.562 - 0.826i)

Particular Values

L(1)L(1) ≈\approx 1.2809656941.280965694
L(12)L(\frac12) ≈\approx 1.2809656941.280965694
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 1
3 1 1
7 1+(1.44+2.21i)T 1 + (1.44 + 2.21i)T
good5 1+(0.310−0.537i)T+(−2.5−4.33i)T2 1 + (0.310 - 0.537i)T + (-2.5 - 4.33i)T^{2}
11 1+(1.91−1.10i)T+(5.5−9.52i)T2 1 + (1.91 - 1.10i)T + (5.5 - 9.52i)T^{2}
13 1−0.963iT−13T2 1 - 0.963iT - 13T^{2}
17 1+(−0.653−1.13i)T+(−8.5+14.7i)T2 1 + (-0.653 - 1.13i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.57+0.911i)T+(9.5+16.4i)T2 1 + (1.57 + 0.911i)T + (9.5 + 16.4i)T^{2}
23 1+(−5.27−3.04i)T+(11.5+19.9i)T2 1 + (-5.27 - 3.04i)T + (11.5 + 19.9i)T^{2}
29 1−6.68iT−29T2 1 - 6.68iT - 29T^{2}
31 1+(−3.48+2.01i)T+(15.5−26.8i)T2 1 + (-3.48 + 2.01i)T + (15.5 - 26.8i)T^{2}
37 1+(−0.165+0.286i)T+(−18.5−32.0i)T2 1 + (-0.165 + 0.286i)T + (-18.5 - 32.0i)T^{2}
41 1−11.7T+41T2 1 - 11.7T + 41T^{2}
43 1+2.18T+43T2 1 + 2.18T + 43T^{2}
47 1+(1.27−2.20i)T+(−23.5−40.7i)T2 1 + (1.27 - 2.20i)T + (-23.5 - 40.7i)T^{2}
53 1+(−0.985+0.569i)T+(26.5−45.8i)T2 1 + (-0.985 + 0.569i)T + (26.5 - 45.8i)T^{2}
59 1+(−6.20−10.7i)T+(−29.5+51.0i)T2 1 + (-6.20 - 10.7i)T + (-29.5 + 51.0i)T^{2}
61 1+(−0.603−0.348i)T+(30.5+52.8i)T2 1 + (-0.603 - 0.348i)T + (30.5 + 52.8i)T^{2}
67 1+(3.61+6.25i)T+(−33.5+58.0i)T2 1 + (3.61 + 6.25i)T + (-33.5 + 58.0i)T^{2}
71 1−7.40iT−71T2 1 - 7.40iT - 71T^{2}
73 1+(2.17−1.25i)T+(36.5−63.2i)T2 1 + (2.17 - 1.25i)T + (36.5 - 63.2i)T^{2}
79 1+(−4.52+7.83i)T+(−39.5−68.4i)T2 1 + (-4.52 + 7.83i)T + (-39.5 - 68.4i)T^{2}
83 1−8.85T+83T2 1 - 8.85T + 83T^{2}
89 1+(4.20−7.29i)T+(−44.5−77.0i)T2 1 + (4.20 - 7.29i)T + (-44.5 - 77.0i)T^{2}
97 1−12.4iT−97T2 1 - 12.4iT - 97T^{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

−9.560050896761407919446574184718, −8.939731305351096913318369641953, −7.80077840701541208751959642316, −7.19914196339591648034661148173, −6.53065945742351604049675765866, −5.44286516128832762601182208795, −4.51653513764435873193846852627, −3.56539094117109623514778721157, −2.66823649793337296276859810215, −1.13568785292751868095358800419, 0.58512265967090040623169241082, 2.37546869996277260073052503565, 3.09150311924977690435047122635, 4.35360590919092273018400808219, 5.25125429954569123120304246373, 6.06505872703600359341276182556, 6.81721673417968216360257163884, 7.956868164028889887016063385595, 8.512390346242674387545149647083, 9.308191977157950065148828971117

Graph of the ZZ-function along the critical line