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Properties

Label 2-756-28.3-c1-0-27
Degree $2$
Conductor $756$
Sign $0.555 - 0.831i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
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  + (1.41 + 0.0418i)2-s + (1.99 + 0.118i)4-s + (−1.51 + 0.872i)5-s + (1.00 + 2.44i)7-s + (2.81 + 0.250i)8-s + (−2.17 + 1.17i)10-s + (−2.72 − 1.57i)11-s + 3.39i·13-s + (1.31 + 3.50i)14-s + (3.97 + 0.472i)16-s + (3.59 + 2.07i)17-s + (2.67 + 4.62i)19-s + (−3.12 + 1.56i)20-s + (−3.78 − 2.33i)22-s + (7.05 − 4.07i)23-s + ⋯
L(s)  = 1  + (0.999 + 0.0295i)2-s + (0.998 + 0.0591i)4-s + (−0.675 + 0.390i)5-s + (0.380 + 0.924i)7-s + (0.996 + 0.0886i)8-s + (−0.687 + 0.370i)10-s + (−0.822 − 0.474i)11-s + 0.942i·13-s + (0.352 + 0.935i)14-s + (0.992 + 0.118i)16-s + (0.871 + 0.503i)17-s + (0.613 + 1.06i)19-s + (−0.697 + 0.349i)20-s + (−0.807 − 0.498i)22-s + (1.47 − 0.849i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.555 - 0.831i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.555 - 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.34220 + 1.25115i\)
\(L(\frac12)\) \(\approx\) \(2.34220 + 1.25115i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0418i)T \)
3 \( 1 \)
7 \( 1 + (-1.00 - 2.44i)T \)
good5 \( 1 + (1.51 - 0.872i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.72 + 1.57i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.39iT - 13T^{2} \)
17 \( 1 + (-3.59 - 2.07i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.67 - 4.62i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.05 + 4.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.25T + 29T^{2} \)
31 \( 1 + (-1.20 + 2.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.00 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.0436iT - 41T^{2} \)
43 \( 1 + 5.38iT - 43T^{2} \)
47 \( 1 + (1.52 + 2.64i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.03 - 3.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.89 + 6.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.01 - 0.588i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.53 - 4.35i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.7iT - 71T^{2} \)
73 \( 1 + (-14.4 - 8.36i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.56 + 1.48i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.83T + 83T^{2} \)
89 \( 1 + (4.07 - 2.35i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.9iT - 97T^{2} \)
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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

−10.97750175416694760097172481605, −9.751760607525487959806133408903, −8.531710380268584432040615199327, −7.73667662214902453423912135293, −6.95697523779438156721908041534, −5.75613632225536369466851041155, −5.23825239569022776316047136762, −3.94573405365850107151593668786, −3.11537820434167385669654729621, −1.90428395685598856866122259197, 1.07107065477319206101844240712, 2.86460211095209251760207022668, 3.73637300633917462670412640668, 5.00172557733311713021913740809, 5.20381735126266559362061308115, 6.82232492973118421035250644073, 7.58200057974068239863853938002, 8.023410725824832493492148051445, 9.567016548648615199396042566991, 10.46948965391565052268491292061

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