Properties

Label 2-2736-57.50-c1-0-9
Degree $2$
Conductor $2736$
Sign $-0.766 - 0.641i$
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  + (−3.15 + 1.82i)5-s − 1.21·7-s + 3.18i·11-s + (3.24 + 1.87i)13-s + (5.49 − 3.17i)17-s + (−1.51 − 4.08i)19-s + (8.02 + 4.63i)23-s + (4.15 − 7.20i)25-s + (−2.76 + 4.79i)29-s − 7.28i·31-s + (3.84 − 2.22i)35-s + 7.63i·37-s + (5.22 + 9.04i)41-s + (2.58 + 4.47i)43-s + (−9.64 − 5.56i)47-s + ⋯
L(s)  = 1  + (−1.41 + 0.815i)5-s − 0.460·7-s + 0.961i·11-s + (0.899 + 0.519i)13-s + (1.33 − 0.769i)17-s + (−0.348 − 0.937i)19-s + (1.67 + 0.965i)23-s + (0.831 − 1.44i)25-s + (−0.513 + 0.889i)29-s − 1.30i·31-s + (0.650 − 0.375i)35-s + 1.25i·37-s + (0.815 + 1.41i)41-s + (0.393 + 0.681i)43-s + (−1.40 − 0.811i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.766 - 0.641i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.766 - 0.641i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8896714676\)
\(L(\frac12)\) \(\approx\) \(0.8896714676\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.51 + 4.08i)T \)
good5 \( 1 + (3.15 - 1.82i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.21T + 7T^{2} \)
11 \( 1 - 3.18iT - 11T^{2} \)
13 \( 1 + (-3.24 - 1.87i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-5.49 + 3.17i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-8.02 - 4.63i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.76 - 4.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.28iT - 31T^{2} \)
37 \( 1 - 7.63iT - 37T^{2} \)
41 \( 1 + (-5.22 - 9.04i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.58 - 4.47i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (9.64 + 5.56i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.30 - 10.9i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.27 + 7.40i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.41 + 2.45i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.06 + 3.50i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.96 - 3.39i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.32 + 2.29i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.33 - 4.23i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.122iT - 83T^{2} \)
89 \( 1 + (5.93 - 10.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.33 - 4.23i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.306305930285783442206305001933, −8.121605320297111371632900058939, −7.56295579923598457336531598033, −6.93850591072152020974642646303, −6.34392390693602034517681863704, −5.05478860078453229206092714142, −4.33788368209346362499767238523, −3.31813940949136584228244146106, −2.93772516872523427743082122774, −1.25043390292138805069141564276, 0.34852035853642008565442169298, 1.31131486163877452296161314748, 3.19730185950320997709035744642, 3.58775097310568135584193734949, 4.44213164907589338426131953590, 5.51043833470875520849256265452, 6.07919348057126287327311310468, 7.19405709293101069939052569976, 7.939180500835250490991520271359, 8.526780902753487310953608100528

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