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Properties

Label 2-2736-19.11-c1-0-46
Degree $2$
Conductor $2736$
Sign $-0.745 + 0.666i$
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  + (1.53 − 2.65i)5-s + 2.06·7-s − 6.45·11-s + (−0.5 − 0.866i)13-s + (0.694 − 1.20i)17-s + (3.75 + 2.20i)19-s + (−1.53 − 2.65i)23-s + (−2.19 − 3.80i)25-s + (−1.75 − 3.04i)29-s − 9.45·31-s + (3.16 − 5.47i)35-s − 2.38·37-s + (5.06 − 8.77i)41-s + (3.03 − 5.25i)43-s + (3 + 5.19i)47-s + ⋯
L(s)  = 1  + (0.685 − 1.18i)5-s + 0.780·7-s − 1.94·11-s + (−0.138 − 0.240i)13-s + (0.168 − 0.291i)17-s + (0.862 + 0.506i)19-s + (−0.319 − 0.553i)23-s + (−0.438 − 0.760i)25-s + (−0.326 − 0.565i)29-s − 1.69·31-s + (0.534 − 0.925i)35-s − 0.392·37-s + (0.790 − 1.36i)41-s + (0.462 − 0.800i)43-s + (0.437 + 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.745 + 0.666i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.745 + 0.666i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.338846714\)
\(L(\frac12)\) \(\approx\) \(1.338846714\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.75 - 2.20i)T \)
good5 \( 1 + (-1.53 + 2.65i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 2.06T + 7T^{2} \)
11 \( 1 + 6.45T + 11T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.694 + 1.20i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.53 + 2.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.75 + 3.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.45T + 31T^{2} \)
37 \( 1 + 2.38T + 37T^{2} \)
41 \( 1 + (-5.06 + 8.77i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.03 + 5.25i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.29 + 9.16i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.59 + 9.69i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.56 + 4.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.72 - 2.99i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.36 - 5.83i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.56 + 7.90i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.790 - 1.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 17.6T + 83T^{2} \)
89 \( 1 + (-5.22 - 9.05i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.36 - 5.83i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.445633787158182339744377106477, −7.87541492982159249272708258477, −7.29141211678106810752792656466, −5.85542969288950305630603553427, −5.25377574671503405890138448738, −5.01123757203987414258097764502, −3.79945111775864957728925958966, −2.50551261318894367149633627038, −1.71993984542851031793562653439, −0.39763673163171008285956838621, 1.62316654337999807649646774641, 2.58562963904460161575548206957, 3.18912488294597481249411255858, 4.51135372169023633209771583343, 5.45886977550301574198817007663, 5.82350642010608993000409310148, 7.06262477129720483939400944494, 7.50120399318009221338520105360, 8.171452717504986188172816869315, 9.246840196919984396239793551515

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