Properties

Label 2-2736-1.1-c1-0-21
Degree $2$
Conductor $2736$
Sign $1$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s + 3·7-s − 11-s − 2·13-s + 5·17-s − 19-s − 4·23-s + 4·25-s + 6·29-s + 2·31-s + 9·35-s + 8·37-s + 8·41-s − 13·43-s + 13·47-s + 2·49-s + 6·53-s − 3·55-s + 4·59-s − 13·61-s − 6·65-s − 4·67-s − 8·71-s − 3·73-s − 3·77-s + 4·79-s + 4·83-s + ⋯
L(s)  = 1  + 1.34·5-s + 1.13·7-s − 0.301·11-s − 0.554·13-s + 1.21·17-s − 0.229·19-s − 0.834·23-s + 4/5·25-s + 1.11·29-s + 0.359·31-s + 1.52·35-s + 1.31·37-s + 1.24·41-s − 1.98·43-s + 1.89·47-s + 2/7·49-s + 0.824·53-s − 0.404·55-s + 0.520·59-s − 1.66·61-s − 0.744·65-s − 0.488·67-s − 0.949·71-s − 0.351·73-s − 0.341·77-s + 0.450·79-s + 0.439·83-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.778036672\)
\(L(\frac12)\) \(\approx\) \(2.778036672\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good5 \( 1 - 3 T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.840684801067866495816084570268, −8.008897510078389419450453308607, −7.48067865026855156296833251878, −6.33474833986742060792549004842, −5.72433580273021334484064561531, −5.03108925927996165499375395008, −4.26173838819332071548971706739, −2.85330315828936229927285829873, −2.08055958925744355984105145505, −1.13058059718558844101553025407, 1.13058059718558844101553025407, 2.08055958925744355984105145505, 2.85330315828936229927285829873, 4.26173838819332071548971706739, 5.03108925927996165499375395008, 5.72433580273021334484064561531, 6.33474833986742060792549004842, 7.48067865026855156296833251878, 8.008897510078389419450453308607, 8.840684801067866495816084570268

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