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Properties

Label 2-2736-228.83-c1-0-39
Degree $2$
Conductor $2736$
Sign $-0.986 + 0.164i$
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  + (3.37 − 1.95i)5-s − 3.27i·7-s − 5.67·11-s + (−2.34 + 4.06i)13-s + (1.20 − 0.695i)17-s + (−3.88 + 1.97i)19-s + (−3.59 + 6.22i)23-s + (5.10 − 8.84i)25-s + (−4.52 − 2.61i)29-s − 9.36i·31-s + (−6.38 − 11.0i)35-s − 8.48·37-s + (3.32 − 1.91i)41-s + (−0.584 + 0.337i)43-s + (3.54 − 6.14i)47-s + ⋯
L(s)  = 1  + (1.51 − 0.872i)5-s − 1.23i·7-s − 1.71·11-s + (−0.651 + 1.12i)13-s + (0.292 − 0.168i)17-s + (−0.891 + 0.453i)19-s + (−0.749 + 1.29i)23-s + (1.02 − 1.76i)25-s + (−0.840 − 0.485i)29-s − 1.68i·31-s + (−1.07 − 1.87i)35-s − 1.39·37-s + (0.519 − 0.299i)41-s + (−0.0891 + 0.0514i)43-s + (0.517 − 0.896i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.986 + 0.164i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.986 + 0.164i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8941029446\)
\(L(\frac12)\) \(\approx\) \(0.8941029446\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.88 - 1.97i)T \)
good5 \( 1 + (-3.37 + 1.95i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.27iT - 7T^{2} \)
11 \( 1 + 5.67T + 11T^{2} \)
13 \( 1 + (2.34 - 4.06i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.20 + 0.695i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.59 - 6.22i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.52 + 2.61i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.36iT - 31T^{2} \)
37 \( 1 + 8.48T + 37T^{2} \)
41 \( 1 + (-3.32 + 1.91i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.584 - 0.337i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.54 + 6.14i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.58 + 2.64i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.63 + 9.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.574 + 0.994i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.96 - 3.44i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.92 + 8.52i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.95 - 10.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.25 - 3.61i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.05T + 83T^{2} \)
89 \( 1 + (13.8 + 7.97i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.66 - 8.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.434348258220663860622695360061, −7.71109415395804575823361888767, −7.01587860953649882027232062600, −5.96160127065218785704469002679, −5.41032272798863272142470779668, −4.62963989780453205414983394833, −3.78915745194588702191218237737, −2.27607351405898066251183984678, −1.75856597407711000407466247947, −0.24472319049569255102427485485, 1.91616929238036067075587992273, 2.65264556356541152632420070254, 3.00929281498958056157651622389, 4.85546127946311504236497934625, 5.49087376810051440099202791705, 5.92908999691405860476378291489, 6.78269363152652059148915216232, 7.67852457287908345513678949525, 8.522765685334027746150597978218, 9.168903088581284482886393874626

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