Properties

Label 2-2736-76.43-c0-0-1
Degree $2$
Conductor $2736$
Sign $0.672 + 0.740i$
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  + (1.70 − 0.984i)7-s + (−0.266 − 1.50i)13-s + (0.5 + 0.866i)19-s + (−0.173 − 0.984i)25-s + (−0.592 + 0.342i)31-s − 1.87·37-s + (−0.439 + 0.524i)43-s + (1.43 − 2.49i)49-s + (1.17 − 0.984i)61-s + (−0.673 + 1.85i)67-s + (−0.266 + 1.50i)73-s + (1.93 + 0.342i)79-s + (−1.93 − 2.31i)91-s + (−0.939 + 0.342i)97-s + (1.11 + 0.642i)103-s + ⋯
L(s)  = 1  + (1.70 − 0.984i)7-s + (−0.266 − 1.50i)13-s + (0.5 + 0.866i)19-s + (−0.173 − 0.984i)25-s + (−0.592 + 0.342i)31-s − 1.87·37-s + (−0.439 + 0.524i)43-s + (1.43 − 2.49i)49-s + (1.17 − 0.984i)61-s + (−0.673 + 1.85i)67-s + (−0.266 + 1.50i)73-s + (1.93 + 0.342i)79-s + (−1.93 − 2.31i)91-s + (−0.939 + 0.342i)97-s + (1.11 + 0.642i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.672 + 0.740i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :0),\ 0.672 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.445135943\)
\(L(\frac12)\) \(\approx\) \(1.445135943\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (-1.70 + 0.984i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.592 - 0.342i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.87T + T^{2} \)
41 \( 1 + (-0.939 - 0.342i)T^{2} \)
43 \( 1 + (0.439 - 0.524i)T + (-0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 + (0.173 - 0.984i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (0.673 - 1.85i)T + (-0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (-1.93 - 0.342i)T + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.939 + 0.342i)T^{2} \)
97 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
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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

−8.578842431067119737246515234921, −8.123532973846039146538908869811, −7.55790541402751341260690301822, −6.81653154910947266112633030739, −5.53014926531502327123002378274, −5.12627693218936796123160523509, −4.17792939747755675968107813779, −3.35347556209531539868810905284, −2.04753858091819288430427177756, −1.02134320580787487600268775313, 1.65383534467245632856282335117, 2.18261228895643057486229214107, 3.50761405254607011434148827038, 4.67026880240693195162183169745, 5.06543754003801735963696710384, 5.90762812265530115618272991299, 7.01456553909952176897685990955, 7.54138612005558656251061701992, 8.537354226016841855303327643472, 8.971732591742995713266219319200

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