Properties

Label 2-2736-76.75-c1-0-47
Degree $2$
Conductor $2736$
Sign $-0.917 - 0.397i$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
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  − 3·5-s − 1.73i·7-s − 5.19i·11-s − 6.92i·13-s − 3·17-s + (4 + 1.73i)19-s + 3.46i·23-s + 4·25-s − 4·31-s + 5.19i·35-s + 6.92i·37-s − 6.92i·41-s − 8.66i·43-s + 8.66i·47-s + 4·49-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.654i·7-s − 1.56i·11-s − 1.92i·13-s − 0.727·17-s + (0.917 + 0.397i)19-s + 0.722i·23-s + 0.800·25-s − 0.718·31-s + 0.878i·35-s + 1.13i·37-s − 1.08i·41-s − 1.32i·43-s + 1.26i·47-s + 0.571·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.917 - 0.397i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.917 - 0.397i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4034990806\)
\(L(\frac12)\) \(\approx\) \(0.4034990806\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + (-4 - 1.73i)T \)
good5 \( 1 + 3T + 5T^{2} \)
7 \( 1 + 1.73iT - 7T^{2} \)
11 \( 1 + 5.19iT - 11T^{2} \)
13 \( 1 + 6.92iT - 13T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 6.92iT - 37T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 + 8.66iT - 43T^{2} \)
47 \( 1 - 8.66iT - 47T^{2} \)
53 \( 1 - 6.92iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 7T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 5T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−8.288286964029819327067385635200, −7.64507147202340724519181813769, −7.20448437257015446406325909788, −5.93724806769214772529088030741, −5.38686810365043352245533050439, −4.24188192936261500795265067956, −3.44885171456428312088057428451, −3.01558305608033511446890935733, −1.05701477312683819753789372682, −0.15325068011453699940967475640, 1.72354329503954401126436974481, 2.62321827240139958344867851899, 3.93691725114823921101828932161, 4.41426893708808648998914503636, 5.13698122377714035576864543291, 6.46924621122733409412709229248, 7.05976695935073563409145620430, 7.60665540134120069815031270274, 8.558571977932800597808796766576, 9.212291945496614031425153989737

Graph of the $Z$-function along the critical line