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Properties

Label 2-2736-19.11-c1-0-26
Degree $2$
Conductor $2736$
Sign $0.827 - 0.560i$
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  + (−0.795 + 1.37i)5-s + 3.87·7-s + 0.409·11-s + (1.64 + 2.84i)13-s + (2.87 − 4.98i)17-s + (3.43 + 2.68i)19-s + (0.0214 + 0.0372i)23-s + (1.23 + 2.14i)25-s + (−2.69 − 4.66i)29-s − 0.773·31-s + (−3.08 + 5.34i)35-s − 0.547·37-s + (5.57 − 9.65i)41-s + (−3.26 + 5.65i)43-s + (3.28 + 5.69i)47-s + ⋯
L(s)  = 1  + (−0.355 + 0.615i)5-s + 1.46·7-s + 0.123·11-s + (0.455 + 0.788i)13-s + (0.698 − 1.20i)17-s + (0.787 + 0.616i)19-s + (0.00448 + 0.00776i)23-s + (0.247 + 0.428i)25-s + (−0.500 − 0.866i)29-s − 0.138·31-s + (−0.521 + 0.902i)35-s − 0.0899·37-s + (0.870 − 1.50i)41-s + (−0.497 + 0.862i)43-s + (0.479 + 0.830i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.827 - 0.560i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.827 - 0.560i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.232397836\)
\(L(\frac12)\) \(\approx\) \(2.232397836\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.795 - 1.37i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.87T + 7T^{2} \)
11 \( 1 - 0.409T + 11T^{2} \)
13 \( 1 + (-1.64 - 2.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.87 + 4.98i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.0214 - 0.0372i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.69 + 4.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.773T + 31T^{2} \)
37 \( 1 + 0.547T + 37T^{2} \)
41 \( 1 + (-5.57 + 9.65i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.26 - 5.65i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.28 - 5.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.49 + 7.78i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.899 - 1.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.537 + 0.931i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.34 - 2.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.16 - 12.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.68 - 2.90i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.67 + 11.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.3T + 83T^{2} \)
89 \( 1 + (-5.85 - 10.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.02 + 13.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.930231524741262499429036389333, −7.84246756362803589082987627638, −7.61108146525783092609428604731, −6.75767260984771293937494088964, −5.71287156893617605567867234016, −5.00917630452329410574872272693, −4.13275983797748158222581591067, −3.28290786263156172437350488898, −2.14426276207927979775177529322, −1.11374334189806814935399561317, 0.926551944474021563720432999968, 1.74521344065647341391379195909, 3.13458779625966393030711314530, 4.07089499018969386044138625153, 4.92304801068022242002002844931, 5.44929467954476737939584343022, 6.39451754941211954724010237153, 7.61779815252433174957298965694, 7.939348694631860127875435756347, 8.643206348964571436671341904170

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