Properties

Label 2-2736-76.27-c1-0-28
Degree $2$
Conductor $2736$
Sign $0.977 - 0.211i$
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  + 1.73i·7-s + (4.5 − 2.59i)13-s + (−4 − 1.73i)19-s + (2.5 + 4.33i)25-s + 7·31-s − 5.19i·37-s + (1.5 + 0.866i)43-s + 4·49-s + (−0.5 − 0.866i)61-s + (5.5 + 9.52i)67-s + (8.5 − 14.7i)73-s + (−8.5 + 14.7i)79-s + (4.5 + 7.79i)91-s + (12 + 6.92i)97-s − 7·103-s + ⋯
L(s)  = 1  + 0.654i·7-s + (1.24 − 0.720i)13-s + (−0.917 − 0.397i)19-s + (0.5 + 0.866i)25-s + 1.25·31-s − 0.854i·37-s + (0.228 + 0.132i)43-s + 0.571·49-s + (−0.0640 − 0.110i)61-s + (0.671 + 1.16i)67-s + (0.994 − 1.72i)73-s + (−0.956 + 1.65i)79-s + (0.471 + 0.817i)91-s + (1.21 + 0.703i)97-s − 0.689·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.977 - 0.211i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.977 - 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.900212836\)
\(L(\frac12)\) \(\approx\) \(1.900212836\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 1.73iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-4.5 + 2.59i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 + 5.19iT - 37T^{2} \)
41 \( 1 + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.5 - 0.866i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.5 + 14.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.5 - 14.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12 - 6.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.719140888055867441608011788356, −8.306474864771470966562884629388, −7.35411392181660839900008508374, −6.43428391563242308233867846327, −5.83454040499677930876511298497, −5.03376896420342561840532107718, −4.03817414099960516379896148426, −3.13815437477491998902486821030, −2.20334430234549329988242126216, −0.910470052617604753593672075841, 0.855825255866622988987925164973, 1.98287532879091870427986315871, 3.19214276063401651569658488838, 4.12763693954251826797621201517, 4.65295640390820777370049660403, 5.93118469062646253544049786026, 6.48565859301277859419015297691, 7.18380769215877349521827336958, 8.295281912548680510010605338664, 8.551437021430137707522251499780

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