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Properties

Label 2-2736-57.50-c1-0-10
Degree $2$
Conductor $2736$
Sign $-0.310 - 0.950i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.310 - 0.950i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.310 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9352212702\)
\(L(\frac12)\) \(\approx\) \(0.9352212702\)
\(L(\frac{3}{2})\) 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

−8.775487955521134267370718948771, −8.240022608951992960242019205031, −7.60330113971815300074325408408, −6.83333050818752001256074266515, −6.16756381495474996627116508060, −5.13237565769288223967723997842, −3.97999228336350007480252722058, −3.54120441736843278114388965326, −2.74247721537555785260852267136, −1.06224519409815560540637027866, 0.37846828283408339986105715899, 1.53665107665144134745113944607, 3.06329581186920185313893485558, 3.87258802611920957156462395681, 4.71465159425668124353460667074, 5.13286823687706660019402289438, 6.42426074131210483025733295249, 7.39687974773898281258739220602, 7.73100098867380486695984296595, 8.605722182378583027009083013642

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