Properties

Label 2-2736-76.11-c0-0-0
Degree $2$
Conductor $2736$
Sign $0.305 - 0.952i$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·7-s + (−0.5 − 0.866i)13-s + 19-s + (0.5 + 0.866i)25-s + 1.73i·31-s − 37-s + (1.5 + 0.866i)43-s − 1.99·49-s + (0.5 + 0.866i)61-s + (−1.5 + 0.866i)67-s + (−0.5 + 0.866i)73-s + (−1.5 − 0.866i)79-s + (1.49 − 0.866i)91-s + (1 − 1.73i)97-s + 1.73i·103-s + ⋯
L(s)  = 1  + 1.73i·7-s + (−0.5 − 0.866i)13-s + 19-s + (0.5 + 0.866i)25-s + 1.73i·31-s − 37-s + (1.5 + 0.866i)43-s − 1.99·49-s + (0.5 + 0.866i)61-s + (−1.5 + 0.866i)67-s + (−0.5 + 0.866i)73-s + (−1.5 − 0.866i)79-s + (1.49 − 0.866i)91-s + (1 − 1.73i)97-s + 1.73i·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.305 - 0.952i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (847, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :0),\ 0.305 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.138800854\)
\(L(\frac12)\) \(\approx\) \(1.138800854\)
\(L(1)\) 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 \( 1 - T \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 - 1.73iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - 1.73iT - T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.952910772874176106280888074193, −8.655110398597480460679181888037, −7.63823020565407308784191310789, −6.96516226400799493069668125762, −5.79609977516594648501204724341, −5.46945916640562172902027589448, −4.66579629539841551547516813598, −3.18985694617474133857687924156, −2.76213216514709892984274457454, −1.50606362395437365325097187522, 0.76959534355878156740201573646, 2.05880563793082294983593070623, 3.31808992649296495843495233733, 4.17810278864933120319254758079, 4.70815618028868880075134397474, 5.83912396319886297117849941603, 6.77888406630811769064961417941, 7.34073364062009789488295651085, 7.87273262613030846417016401448, 8.943994934304339261617236592804

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