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Properties

Label 2-2736-76.31-c1-0-36
Degree $2$
Conductor $2736$
Sign $-0.0279 + 0.999i$
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  + (0.675 + 1.17i)5-s + 1.45i·7-s − 3.18i·11-s + (−2.23 − 1.28i)13-s + (−2.08 − 3.61i)17-s + (2.43 + 3.61i)19-s + (−6.49 − 3.75i)23-s + (1.58 − 2.74i)25-s + (0.734 + 0.423i)29-s − 0.351·31-s + (−1.70 + 0.985i)35-s − 6.89i·37-s + (4.05 − 2.34i)41-s + (−6.52 + 3.76i)43-s + (−8.04 − 4.64i)47-s + ⋯
L(s)  = 1  + (0.302 + 0.523i)5-s + 0.550i·7-s − 0.961i·11-s + (−0.619 − 0.357i)13-s + (−0.505 − 0.876i)17-s + (0.559 + 0.828i)19-s + (−1.35 − 0.782i)23-s + (0.317 − 0.549i)25-s + (0.136 + 0.0787i)29-s − 0.0632·31-s + (−0.288 + 0.166i)35-s − 1.13i·37-s + (0.633 − 0.365i)41-s + (−0.995 + 0.574i)43-s + (−1.17 − 0.677i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0279 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0279 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.0279 + 0.999i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1855, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.0279 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.128942898\)
\(L(\frac12)\) \(\approx\) \(1.128942898\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.43 - 3.61i)T \)
good5 \( 1 + (-0.675 - 1.17i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 1.45iT - 7T^{2} \)
11 \( 1 + 3.18iT - 11T^{2} \)
13 \( 1 + (2.23 + 1.28i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.08 + 3.61i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (6.49 + 3.75i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.734 - 0.423i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.351T + 31T^{2} \)
37 \( 1 + 6.89iT - 37T^{2} \)
41 \( 1 + (-4.05 + 2.34i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.52 - 3.76i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.04 + 4.64i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.76 + 5.05i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.675 + 1.17i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.29 + 7.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.08 + 1.88i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.438 + 0.758i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.67 - 11.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.52 - 4.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.22iT - 83T^{2} \)
89 \( 1 + (-3.81 - 2.20i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.79 - 4.49i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.488132123865254145918797193912, −8.035153643154046528108387279968, −7.02898248328836693804683664748, −6.28933617368541400095114332834, −5.62989410734709097577464306180, −4.82761545894111421561918716002, −3.68187050985239033307522051692, −2.80586568202150164814110028928, −2.03826162957323264810365632690, −0.35746399625445419638985699504, 1.32716687998824924181914844926, 2.19810311902091220066284625256, 3.45421718624740848874046114061, 4.47360193964059414280957942152, 4.92692531765117472375253247132, 5.97319247542928932757522667353, 6.81052690989917705677011997946, 7.48868224412554422815348189127, 8.210791695510137811673874281913, 9.165892462270150766374365965615

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