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Properties

Label 2-2736-57.50-c1-0-25
Degree $2$
Conductor $2736$
Sign $0.999 + 0.0242i$
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  + (3.57 − 2.06i)5-s + 0.233·7-s + 3.83i·11-s + (2.59 + 1.49i)13-s + (0.815 − 0.470i)17-s + (3.73 + 2.24i)19-s + (−1.80 − 1.04i)23-s + (6.03 − 10.4i)25-s + (−0.168 + 0.291i)29-s + 0.259i·31-s + (0.836 − 0.482i)35-s + 2.86i·37-s + (4.65 + 8.06i)41-s + (4.13 + 7.16i)43-s + (1.21 + 0.703i)47-s + ⋯
L(s)  = 1  + (1.60 − 0.923i)5-s + 0.0883·7-s + 1.15i·11-s + (0.720 + 0.415i)13-s + (0.197 − 0.114i)17-s + (0.856 + 0.515i)19-s + (−0.376 − 0.217i)23-s + (1.20 − 2.09i)25-s + (−0.0312 + 0.0542i)29-s + 0.0465i·31-s + (0.141 − 0.0816i)35-s + 0.470i·37-s + (0.727 + 1.25i)41-s + (0.630 + 1.09i)43-s + (0.177 + 0.102i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.999 + 0.0242i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.999 + 0.0242i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.743744270\)
\(L(\frac12)\) \(\approx\) \(2.743744270\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.73 - 2.24i)T \)
good5 \( 1 + (-3.57 + 2.06i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 0.233T + 7T^{2} \)
11 \( 1 - 3.83iT - 11T^{2} \)
13 \( 1 + (-2.59 - 1.49i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.815 + 0.470i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.80 + 1.04i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.168 - 0.291i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.259iT - 31T^{2} \)
37 \( 1 - 2.86iT - 37T^{2} \)
41 \( 1 + (-4.65 - 8.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.13 - 7.16i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.21 - 0.703i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.843 + 1.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.25 - 2.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.69 - 2.93i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.78 - 3.34i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.36 + 12.7i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.26 + 2.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.48 - 5.47i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.0iT - 83T^{2} \)
89 \( 1 + (-7.36 + 12.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.9 + 6.89i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.964272028510447063099717819932, −8.191046921216623765068812858225, −7.30143162141083646047568157379, −6.27144961219336181039901794070, −5.83984207933628450882993719324, −4.89165418997434398570737313077, −4.35949654965076702990390742550, −2.94553049344746351100577118463, −1.84206282955098073307477519495, −1.24872527802144282607840895464, 1.03793163926103233437462192521, 2.20734747807260389004051491013, 3.02596527691671693180736406491, 3.81633571408795132003433621846, 5.43208660675983968830321613550, 5.64124797165908481528388826351, 6.43716796214969808500786429287, 7.16155173203141219949882221726, 8.104620241727717111198519965638, 9.005616655539450828962036265626

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