Properties

Label 2-2736-19.11-c1-0-24
Degree $2$
Conductor $2736$
Sign $0.960 - 0.279i$
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  + (0.675 − 1.17i)5-s − 0.351·7-s + 5.52·11-s + (2.58 + 4.47i)13-s + (2.43 + 3.61i)19-s + (−4.41 − 7.63i)23-s + (1.58 + 2.74i)25-s + (1.35 + 2.34i)29-s + 0.524·31-s + (−0.237 + 0.412i)35-s − 37-s + (−1.35 + 2.34i)41-s + (−3.26 + 5.65i)43-s + (3 + 5.19i)47-s − 6.87·49-s + ⋯
L(s)  = 1  + (0.302 − 0.523i)5-s − 0.133·7-s + 1.66·11-s + (0.717 + 1.24i)13-s + (0.559 + 0.828i)19-s + (−0.919 − 1.59i)23-s + (0.317 + 0.549i)25-s + (0.251 + 0.434i)29-s + 0.0941·31-s + (−0.0402 + 0.0696i)35-s − 0.164·37-s + (−0.211 + 0.365i)41-s + (−0.497 + 0.861i)43-s + (0.437 + 0.757i)47-s − 0.982·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.960 - 0.279i$
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: \((2,\ 2736,\ (\ :1/2),\ 0.960 - 0.279i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.203737900\)
\(L(\frac12)\) \(\approx\) \(2.203737900\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.43 - 3.61i)T \)
good5 \( 1 + (-0.675 + 1.17i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.351T + 7T^{2} \)
11 \( 1 - 5.52T + 11T^{2} \)
13 \( 1 + (-2.58 - 4.47i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.41 + 7.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.35 - 2.34i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.524T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (1.35 - 2.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.26 - 5.65i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.02 + 3.51i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.76 + 4.78i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.938 - 1.62i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.99 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.52 - 4.37i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.85 - 6.67i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.91 + 6.77i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.34T + 83T^{2} \)
89 \( 1 + (2.32 + 4.02i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.90 + 11.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.761286327919115421582133707399, −8.431231473674499456115262439379, −7.20218988329194563035885250047, −6.42534533597118403682796811298, −6.03785890373256351676224762977, −4.80333354694580037386765488111, −4.13996704940570783161401196729, −3.34070918428583191638113892583, −1.87622345109952783893745608490, −1.15393194193708779487454245128, 0.863744173324517766121023486179, 2.02448728082545304952933696029, 3.28754615230875397822875563933, 3.74526929174479374753780484246, 4.94971718631925638369589255222, 5.89032289552005685292841042392, 6.42905433914185723791077243532, 7.20487772420519105143122006657, 8.029518094608214744209174579190, 8.870791927417131105574580358075

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