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Properties

Label 2-2736-57.50-c1-0-22
Degree $2$
Conductor $2736$
Sign $0.331 + 0.943i$
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  + (−1.76 + 1.01i)5-s − 3.69·7-s − 0.605i·11-s + (2.65 + 1.53i)13-s + (−2.40 + 1.39i)17-s + (−0.780 + 4.28i)19-s + (−2.23 − 1.29i)23-s + (−0.422 + 0.732i)25-s + (1.70 − 2.95i)29-s − 4.42i·31-s + (6.52 − 3.76i)35-s + 3.34i·37-s + (1.50 + 2.60i)41-s + (−0.562 − 0.974i)43-s + (−3.05 − 1.76i)47-s + ⋯
L(s)  = 1  + (−0.789 + 0.455i)5-s − 1.39·7-s − 0.182i·11-s + (0.735 + 0.424i)13-s + (−0.584 + 0.337i)17-s + (−0.179 + 0.983i)19-s + (−0.466 − 0.269i)23-s + (−0.0845 + 0.146i)25-s + (0.316 − 0.548i)29-s − 0.795i·31-s + (1.10 − 0.637i)35-s + 0.550i·37-s + (0.234 + 0.406i)41-s + (−0.0858 − 0.148i)43-s + (−0.445 − 0.257i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.331 + 0.943i$
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: \((2,\ 2736,\ (\ :1/2),\ 0.331 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6196507796\)
\(L(\frac12)\) \(\approx\) \(0.6196507796\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.780 - 4.28i)T \)
good5 \( 1 + (1.76 - 1.01i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.69T + 7T^{2} \)
11 \( 1 + 0.605iT - 11T^{2} \)
13 \( 1 + (-2.65 - 1.53i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.40 - 1.39i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.23 + 1.29i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.70 + 2.95i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.42iT - 31T^{2} \)
37 \( 1 - 3.34iT - 37T^{2} \)
41 \( 1 + (-1.50 - 2.60i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.562 + 0.974i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.05 + 1.76i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.984 + 1.70i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.48 + 6.03i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.98 + 8.62i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.324 + 0.187i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.03 + 13.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-8.09 - 14.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.44 + 4.30i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.17iT - 83T^{2} \)
89 \( 1 + (-3.91 + 6.77i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.57 + 4.95i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.564014000804127564171856120219, −7.984165012181122501875939716798, −7.07149223306137156523202517828, −6.32047662233025737590683016927, −5.95470256933152068910406117525, −4.51160407085503272236214718141, −3.69902273561652768517210811835, −3.22300241309575805529489203090, −1.96537329714957331793205805152, −0.26733706383186992936791185551, 0.849781793084375534362156410359, 2.49723273412165989367633621141, 3.40653069317318417088904875306, 4.11586215348943315240144998460, 5.01627317349849563742048669197, 6.01529841696793770306959345628, 6.72266780173853471953362605025, 7.38924332490307734347075758553, 8.335187045774096686379949741965, 8.951918083235552665429812719491

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