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Properties

Label 2-2736-19.7-c1-0-27
Degree $2$
Conductor $2736$
Sign $0.942 + 0.334i$
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  + (1.82 + 3.15i)5-s − 4.64·7-s + 0.354·11-s + (2.14 − 3.71i)13-s + (−1.64 − 2.85i)17-s + (2.64 − 3.46i)19-s + (2.82 − 4.88i)23-s + (−4.14 + 7.18i)25-s + (3.64 − 6.31i)29-s + 0.645·31-s + (−8.46 − 14.6i)35-s − 5·37-s + (2 + 3.46i)41-s + (2.67 + 4.63i)43-s + (−2.64 + 4.58i)47-s + ⋯
L(s)  = 1  + (0.815 + 1.41i)5-s − 1.75·7-s + 0.106·11-s + (0.595 − 1.03i)13-s + (−0.399 − 0.691i)17-s + (0.606 − 0.794i)19-s + (0.588 − 1.01i)23-s + (−0.829 + 1.43i)25-s + (0.676 − 1.17i)29-s + 0.115·31-s + (−1.43 − 2.47i)35-s − 0.821·37-s + (0.312 + 0.541i)41-s + (0.408 + 0.707i)43-s + (−0.385 + 0.668i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.942 + 0.334i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.942 + 0.334i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.586653957\)
\(L(\frac12)\) \(\approx\) \(1.586653957\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 + 3.46i)T \)
good5 \( 1 + (-1.82 - 3.15i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 4.64T + 7T^{2} \)
11 \( 1 - 0.354T + 11T^{2} \)
13 \( 1 + (-2.14 + 3.71i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.64 + 2.85i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.82 + 4.88i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.64 + 6.31i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.645T + 31T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 + (-2 - 3.46i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.67 - 4.63i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.64 - 4.58i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.46 + 6.00i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.82 + 6.62i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.96 + 12.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5 - 8.66i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.14 - 5.44i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.67 - 2.90i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.2T + 83T^{2} \)
89 \( 1 + (-6.82 + 11.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.29 + 9.16i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.046920846930730504617994160194, −7.912038544991486766764636576483, −6.94041287683788699625523278272, −6.49555689469795565887058838702, −6.05595048424363759793799952646, −5.01882235295276316494028019605, −3.64647028854137521624095169115, −2.90388740464471365206284279934, −2.53393378047613483728375356270, −0.60408423926781223813857666800, 1.02889705459625545882793691640, 1.96701204361077581878483796526, 3.33894837675064523450686012690, 4.01227249251537783899558765503, 5.11771102729348820107478581295, 5.81179048132784882460581486983, 6.46084989563486434030339094334, 7.17601844372999939076423165009, 8.422618659018887162586430918451, 9.107992259007859432462715999689

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