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Properties

Label 2-756-63.61-c2-0-0
Degree $2$
Conductor $756$
Sign $-0.760 - 0.649i$
Analytic cond. $20.5995$
Root an. cond. $4.53866$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.08i·5-s + (6.74 + 1.87i)7-s − 15.8·11-s + (−8.96 + 5.17i)13-s + (−20.2 + 11.6i)17-s + (−2.52 − 1.45i)19-s − 6.16·23-s + 15.4·25-s + (−19.5 + 33.7i)29-s + (−0.833 − 0.481i)31-s + (5.79 − 20.7i)35-s + (22.9 − 39.7i)37-s + (29.7 − 17.1i)41-s + (−40.0 + 69.3i)43-s + (−28.9 + 16.7i)47-s + ⋯
L(s)  = 1  − 0.616i·5-s + (0.963 + 0.268i)7-s − 1.43·11-s + (−0.689 + 0.398i)13-s + (−1.19 + 0.688i)17-s + (−0.133 − 0.0768i)19-s − 0.268·23-s + 0.619·25-s + (−0.672 + 1.16i)29-s + (−0.0268 − 0.0155i)31-s + (0.165 − 0.594i)35-s + (0.620 − 1.07i)37-s + (0.725 − 0.418i)41-s + (−0.931 + 1.61i)43-s + (−0.615 + 0.355i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.760 - 0.649i$
Analytic conductor: \(20.5995\)
Root analytic conductor: \(4.53866\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1),\ -0.760 - 0.649i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4984515684\)
\(L(\frac12)\) \(\approx\) \(0.4984515684\)
\(L(2)\) 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 \( 1 \)
7 \( 1 + (-6.74 - 1.87i)T \)
good5 \( 1 + 3.08iT - 25T^{2} \)
11 \( 1 + 15.8T + 121T^{2} \)
13 \( 1 + (8.96 - 5.17i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (20.2 - 11.6i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (2.52 + 1.45i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + 6.16T + 529T^{2} \)
29 \( 1 + (19.5 - 33.7i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (0.833 + 0.481i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-22.9 + 39.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-29.7 + 17.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (40.0 - 69.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (28.9 - 16.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (13.3 + 23.2i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (24.6 + 14.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (82.3 - 47.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (26.8 - 46.4i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 89.3T + 5.04e3T^{2} \)
73 \( 1 + (54.7 - 31.6i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-2.51 - 4.35i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-77.5 - 44.7i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-72.6 - 41.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (57.4 + 33.1i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.70869007250524373406106841278, −9.491689151914192721124983817448, −8.698513254709437916493385308131, −7.996033848497924443861689704606, −7.17090836841653413082340181879, −5.88215023617303991298784962973, −4.95981725161082948851745541490, −4.39224508250404511298893260091, −2.70202639913461777412013098625, −1.65691660794234850826232984776, 0.15866716199916669205982099188, 2.09466680440868728242441311282, 2.97356638526153058203265358107, 4.50308155359237913748518869145, 5.14636550001707938000630838647, 6.32372956140031763937105174555, 7.44220204301933256675369059189, 7.85955012503490345123633450122, 8.901652328896425231933961858294, 10.04439423352511477490892009359

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