Properties

Label 2-2736-76.75-c1-0-4
Degree $2$
Conductor $2736$
Sign $-0.114 - 0.993i$
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  − 2.37·5-s − 2.52i·7-s + 2.52i·11-s − 1.58i·13-s + 0.372·17-s + (−4 − 1.73i)19-s + 1.87i·23-s + 0.627·25-s − 3.16i·29-s + 2.74·31-s + 5.98i·35-s − 1.58i·37-s + 6.92i·41-s + 0.644i·43-s + 0.939i·47-s + ⋯
L(s)  = 1  − 1.06·5-s − 0.954i·7-s + 0.761i·11-s − 0.439i·13-s + 0.0902·17-s + (−0.917 − 0.397i)19-s + 0.391i·23-s + 0.125·25-s − 0.588i·29-s + 0.492·31-s + 1.01i·35-s − 0.260i·37-s + 1.08i·41-s + 0.0983i·43-s + 0.137i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.114 - 0.993i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.114 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6285900663\)
\(L(\frac12)\) \(\approx\) \(0.6285900663\)
\(L(\frac{3}{2})\) 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 \( 1 + (4 + 1.73i)T \)
good5 \( 1 + 2.37T + 5T^{2} \)
7 \( 1 + 2.52iT - 7T^{2} \)
11 \( 1 - 2.52iT - 11T^{2} \)
13 \( 1 + 1.58iT - 13T^{2} \)
17 \( 1 - 0.372T + 17T^{2} \)
23 \( 1 - 1.87iT - 23T^{2} \)
29 \( 1 + 3.16iT - 29T^{2} \)
31 \( 1 - 2.74T + 31T^{2} \)
37 \( 1 + 1.58iT - 37T^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 - 0.644iT - 43T^{2} \)
47 \( 1 - 0.939iT - 47T^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 0.372T + 61T^{2} \)
67 \( 1 + 13.4T + 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 - 13.1T + 73T^{2} \)
79 \( 1 - 6.74T + 79T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 - 13.2iT - 89T^{2} \)
97 \( 1 - 13.2iT - 97T^{2} \)
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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

−9.003638457560240625581793034891, −7.936691265399016940185218364399, −7.70770103100857121984386685090, −6.89035487589820193302034558686, −6.09805229614701810385523546310, −4.85944625468801963236117433567, −4.26170588316308541084313767528, −3.58944344006895952516585196945, −2.46486827481052905037631686302, −1.04070646736043895234988613409, 0.23652182474508181986311652935, 1.86533902966586833474346644876, 2.96247962349701155859142772982, 3.80203283844520693579824779265, 4.60683008674190201247236088569, 5.57910366018975732276592685003, 6.30545369879567739496631235979, 7.13416005820723001972481556817, 8.053641848725363207330039010344, 8.567035649293616778110143142171

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