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Properties

Label 2-1872-13.10-c1-0-6
Degree $2$
Conductor $1872$
Sign $-0.711 - 0.702i$
Analytic cond. $14.9479$
Root an. cond. $3.86626$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $-0.711 - 0.702i$
Analytic conductor: \(14.9479\)
Root analytic conductor: \(3.86626\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1872,\ (\ :1/2),\ -0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.487684791\)
\(L(\frac12)\) \(\approx\) \(1.487684791\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 + (-3.5 - 0.866i)T \)
good5 \( 1 - 3.46iT - 5T^{2} \)
7 \( 1 + (-1.5 - 0.866i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6 + 3.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + (3 + 1.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.5 - 4.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (9 + 5.19i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 - 13.8iT - 83T^{2} \)
89 \( 1 + (6 - 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.5 + 2.59i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line