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Properties

Label 2-2736-76.31-c1-0-28
Degree $2$
Conductor $2736$
Sign $0.536 + 0.844i$
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  + (−1.38 − 2.39i)5-s − 3.26i·7-s + 1.94i·11-s + (3.96 + 2.29i)13-s + (2.27 + 3.94i)17-s + (3.43 − 2.68i)19-s + (7.81 + 4.51i)23-s + (−1.32 + 2.29i)25-s + (−1.78 − 1.03i)29-s + 1.21·31-s + (−7.81 + 4.51i)35-s − 1.73i·37-s + (−3.25 + 1.87i)41-s + (1.82 − 1.05i)43-s + (9.49 + 5.48i)47-s + ⋯
L(s)  = 1  + (−0.618 − 1.07i)5-s − 1.23i·7-s + 0.585i·11-s + (1.10 + 0.635i)13-s + (0.551 + 0.955i)17-s + (0.787 − 0.615i)19-s + (1.62 + 0.940i)23-s + (−0.264 + 0.458i)25-s + (−0.331 − 0.191i)29-s + 0.218·31-s + (−1.32 + 0.762i)35-s − 0.284i·37-s + (−0.508 + 0.293i)41-s + (0.278 − 0.160i)43-s + (1.38 + 0.799i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.536 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.536 + 0.844i$
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: \((2,\ 2736,\ (\ :1/2),\ 0.536 + 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.823535439\)
\(L(\frac12)\) \(\approx\) \(1.823535439\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.43 + 2.68i)T \)
good5 \( 1 + (1.38 + 2.39i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.26iT - 7T^{2} \)
11 \( 1 - 1.94iT - 11T^{2} \)
13 \( 1 + (-3.96 - 2.29i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.27 - 3.94i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-7.81 - 4.51i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.78 + 1.03i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.21T + 31T^{2} \)
37 \( 1 + 1.73iT - 37T^{2} \)
41 \( 1 + (3.25 - 1.87i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.82 + 1.05i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.49 - 5.48i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.40 - 4.27i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.68 + 2.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.322 + 0.559i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.04 + 7.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.13 + 10.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.79 + 6.56i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.04 + 8.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.9iT - 83T^{2} \)
89 \( 1 + (5.93 + 3.42i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (16.4 - 9.47i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.893654104066619460866109297061, −7.75648671125140684203736119380, −7.43161555656803931534463724372, −6.51544802889108053409053770583, −5.48016179497465300213493953631, −4.60552381676295040598187574983, −4.03584750185794633863557161781, −3.26693429598839934928236092155, −1.51465621786021067989322018043, −0.834434599080492803631773302761, 0.991488690823712955980927749286, 2.70411754230602766077363882259, 3.05864061731994452464725885648, 3.96773767915972540722085437974, 5.42702274424383795171645634680, 5.66066872455998977674139246806, 6.81646751749742671386782737602, 7.29357852095342974314872595481, 8.402776388086880521438249352513, 8.678001567131640900936375125998

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