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Properties

Label 2-2736-57.8-c1-0-12
Degree $2$
Conductor $2736$
Sign $0.388 - 0.921i$
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  + (−0.0242 − 0.0140i)5-s + 1.95·7-s + 2.26i·11-s + (−1.04 + 0.605i)13-s + (5.68 + 3.27i)17-s + (−3.90 + 1.93i)19-s + (2.86 − 1.65i)23-s + (−2.49 − 4.32i)25-s + (0.972 + 1.68i)29-s + 7.53i·31-s + (−0.0474 − 0.0274i)35-s − 1.60i·37-s + (5.51 − 9.55i)41-s + (−6.35 + 11.0i)43-s + (−5.51 + 3.18i)47-s + ⋯
L(s)  = 1  + (−0.0108 − 0.00626i)5-s + 0.739·7-s + 0.683i·11-s + (−0.290 + 0.167i)13-s + (1.37 + 0.795i)17-s + (−0.896 + 0.442i)19-s + (0.597 − 0.344i)23-s + (−0.499 − 0.865i)25-s + (0.180 + 0.312i)29-s + 1.35i·31-s + (−0.00802 − 0.00463i)35-s − 0.263i·37-s + (0.861 − 1.49i)41-s + (−0.969 + 1.67i)43-s + (−0.804 + 0.464i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.388 - 0.921i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.388 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.792052236\)
\(L(\frac12)\) \(\approx\) \(1.792052236\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.90 - 1.93i)T \)
good5 \( 1 + (0.0242 + 0.0140i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 1.95T + 7T^{2} \)
11 \( 1 - 2.26iT - 11T^{2} \)
13 \( 1 + (1.04 - 0.605i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-5.68 - 3.27i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.86 + 1.65i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.972 - 1.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.53iT - 31T^{2} \)
37 \( 1 + 1.60iT - 37T^{2} \)
41 \( 1 + (-5.51 + 9.55i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.35 - 11.0i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.51 - 3.18i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.92 - 10.2i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.67 - 4.62i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.233 + 0.403i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.18 + 2.41i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.11 + 1.92i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.85 + 3.21i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.79 - 2.77i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.64iT - 83T^{2} \)
89 \( 1 + (-1.78 - 3.09i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.80 - 5.65i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.871507615627171993328838189373, −8.130919623636551631322747555280, −7.60773960294251260786740234242, −6.68129950553212132968021452403, −5.90283785967745793052272579534, −4.96673300422589026230614323290, −4.34651785422860810257622806625, −3.34219470358104685041511722541, −2.17933184962830651575233453966, −1.26674144546029386539065853195, 0.62154912372056768434992715786, 1.87639785572753738826187556183, 2.98153271739630215319574816439, 3.83392228670572879800360740584, 4.94282930280976064165875947427, 5.43717190575812595438531392777, 6.37750127533369604261589995110, 7.29764098936714638202763957429, 7.968246803116765224807255315709, 8.537595100864674023906679926593

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