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Properties

Label 2-2736-76.31-c1-0-43
Degree $2$
Conductor $2736$
Sign $-0.671 + 0.740i$
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  + (1.5 + 2.59i)5-s − 3.46i·7-s − 3.46i·11-s + (−4.5 − 2.59i)13-s + (1.5 + 2.59i)17-s + (4 − 1.73i)19-s + (−4.5 − 2.59i)23-s + (−2 + 3.46i)25-s + (−7.5 − 4.33i)29-s − 4·31-s + (9 − 5.19i)35-s + (−7.5 + 4.33i)41-s + (−10.5 + 6.06i)43-s + (1.5 + 0.866i)47-s − 4.99·49-s + ⋯
L(s)  = 1  + (0.670 + 1.16i)5-s − 1.30i·7-s − 1.04i·11-s + (−1.24 − 0.720i)13-s + (0.363 + 0.630i)17-s + (0.917 − 0.397i)19-s + (−0.938 − 0.541i)23-s + (−0.400 + 0.692i)25-s + (−1.39 − 0.804i)29-s − 0.718·31-s + (1.52 − 0.878i)35-s + (−1.17 + 0.676i)41-s + (−1.60 + 0.924i)43-s + (0.218 + 0.126i)47-s − 0.714·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.671 + 0.740i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1855, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.671 + 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8889004295\)
\(L(\frac12)\) \(\approx\) \(0.8889004295\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.46iT - 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + (4.5 + 2.59i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (4.5 + 2.59i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.5 + 4.33i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (7.5 - 4.33i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (10.5 - 6.06i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.5 - 0.866i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 + 0.866i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.5 + 7.79i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.5 + 6.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 + (7.5 + 4.33i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.5 + 4.33i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.397370015652534268373341671914, −7.56706297593131693773460330812, −7.17111337240995763810753187466, −6.22259444128984158306333876082, −5.65107159353087039085519855755, −4.57542470381745645237484483860, −3.47619937905243329355179866810, −2.95523602940204372613899049497, −1.74041106135087455574247838880, −0.25843238055216035560896507570, 1.72999439355718123626895946964, 2.10191905586835977574389162155, 3.45567569001419686576063645668, 4.70961742426921954335213273304, 5.31695715163246919843069387004, 5.64095604259584061062358339022, 6.93938064815398689877059183375, 7.54575132955787889438388616072, 8.558300385684690624453017341149, 9.255139258504715353664231124947

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