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Properties

Label 2-1872-13.10-c1-0-6
Degree 22
Conductor 18721872
Sign −0.711−0.702i-0.711 - 0.702i
Analytic cond. 14.947914.9479
Root an. cond. 3.866263.86626
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  + 3.46i·5-s + (1.5 + 0.866i)7-s + (−3 + 1.73i)11-s + (3.5 + 0.866i)13-s + (−3 − 1.73i)19-s + (3 + 5.19i)23-s − 6.99·25-s + (3 + 5.19i)29-s + 1.73i·31-s + (−2.99 + 5.19i)35-s + (6 − 3.46i)41-s + (−0.5 + 0.866i)43-s − 3.46i·47-s + (−2 − 3.46i)49-s − 12·53-s + ⋯
L(s)  = 1  + 1.54i·5-s + (0.566 + 0.327i)7-s + (−0.904 + 0.522i)11-s + (0.970 + 0.240i)13-s + (−0.688 − 0.397i)19-s + (0.625 + 1.08i)23-s − 1.39·25-s + (0.557 + 0.964i)29-s + 0.311i·31-s + (−0.507 + 0.878i)35-s + (0.937 − 0.541i)41-s + (−0.0762 + 0.132i)43-s − 0.505i·47-s + (−0.285 − 0.494i)49-s − 1.64·53-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=((−0.711−0.702i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s+1/2) L(s)=((−0.711−0.702i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: −0.711−0.702i-0.711 - 0.702i
Analytic conductor: 14.947914.9479
Root analytic conductor: 3.866263.86626
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1872(1297,⋅)\chi_{1872} (1297, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1872, ( :1/2), −0.711−0.702i)(2,\ 1872,\ (\ :1/2),\ -0.711 - 0.702i)

Particular Values

L(1)L(1) ≈\approx 1.4876847911.487684791
L(12)L(\frac12) ≈\approx 1.4876847911.487684791
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
13 1+(−3.5−0.866i)T 1 + (-3.5 - 0.866i)T
good5 1−3.46iT−5T2 1 - 3.46iT - 5T^{2}
7 1+(−1.5−0.866i)T+(3.5+6.06i)T2 1 + (-1.5 - 0.866i)T + (3.5 + 6.06i)T^{2}
11 1+(3−1.73i)T+(5.5−9.52i)T2 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2}
17 1+(−8.5−14.7i)T2 1 + (-8.5 - 14.7i)T^{2}
19 1+(3+1.73i)T+(9.5+16.4i)T2 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2}
23 1+(−3−5.19i)T+(−11.5+19.9i)T2 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2}
29 1+(−3−5.19i)T+(−14.5+25.1i)T2 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2}
31 1−1.73iT−31T2 1 - 1.73iT - 31T^{2}
37 1+(18.5−32.0i)T2 1 + (18.5 - 32.0i)T^{2}
41 1+(−6+3.46i)T+(20.5−35.5i)T2 1 + (-6 + 3.46i)T + (20.5 - 35.5i)T^{2}
43 1+(0.5−0.866i)T+(−21.5−37.2i)T2 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2}
47 1+3.46iT−47T2 1 + 3.46iT - 47T^{2}
53 1+12T+53T2 1 + 12T + 53T^{2}
59 1+(3+1.73i)T+(29.5+51.0i)T2 1 + (3 + 1.73i)T + (29.5 + 51.0i)T^{2}
61 1+(0.5−0.866i)T+(−30.5−52.8i)T2 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2}
67 1+(7.5−4.33i)T+(33.5−58.0i)T2 1 + (7.5 - 4.33i)T + (33.5 - 58.0i)T^{2}
71 1+(9+5.19i)T+(35.5+61.4i)T2 1 + (9 + 5.19i)T + (35.5 + 61.4i)T^{2}
73 1+1.73iT−73T2 1 + 1.73iT - 73T^{2}
79 1−11T+79T2 1 - 11T + 79T^{2}
83 1−13.8iT−83T2 1 - 13.8iT - 83T^{2}
89 1+(6−3.46i)T+(44.5−77.0i)T2 1 + (6 - 3.46i)T + (44.5 - 77.0i)T^{2}
97 1+(4.5+2.59i)T+(48.5+84.0i)T2 1 + (4.5 + 2.59i)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

−9.575998133364500977744244022024, −8.692700063809163301382088294009, −7.85368742182022637492133138713, −7.12709110255618987692534609081, −6.48037458057486404128717187484, −5.57727699753901817016429502406, −4.65716645406775715395121822368, −3.48490803466939314477923148091, −2.72693299224450383401279500980, −1.71402981877806298870744535869, 0.55217117108562369530930829237, 1.55261980507020798552462442305, 2.91854700757476451901906785866, 4.28323666195234539858464983190, 4.69179727652537013997358929311, 5.68227205182762207979301500021, 6.32696931124475418608866006642, 7.77246511181238394803567399300, 8.185671717299670862931690995473, 8.761769351219909915723949317334

Graph of the ZZ-function along the critical line