Properties

Label 2-2736-76.75-c1-0-17
Degree $2$
Conductor $2736$
Sign $0.802 - 0.596i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.802 - 0.596i$
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: \((2,\ 2736,\ (\ :1/2),\ 0.802 - 0.596i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.277426738\)
\(L(\frac12)\) \(\approx\) \(1.277426738\)
\(L(\frac{3}{2})\) 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

−8.811150922411199207900730979173, −8.035786314457096910329453818648, −7.67967002013739861012501136935, −6.57884704225468047444702681294, −5.75324240736233862155050319588, −5.11375607271090113218070796303, −3.99496771459034468944252521091, −3.29681036990963501770902268408, −2.37276242559916691270723132503, −0.825027235551973116206661704269, 0.59870720590271626844612130626, 1.90136343353469767999038587950, 3.31217979086211243511308834387, 3.93212728948566397945000114991, 4.68324467876313276239246987529, 5.52340269597874660299111335003, 6.88443820252751587507008177690, 7.16760835549502280031383844642, 7.85453940859785138146928638183, 8.649827278102811512429313810163

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