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Properties

Label 2-756-28.27-c1-0-40
Degree $2$
Conductor $756$
Sign $0.921 - 0.387i$
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  + (1.40 + 0.189i)2-s + (1.92 + 0.532i)4-s + 1.46i·5-s + (2.07 − 1.63i)7-s + (2.60 + 1.11i)8-s + (−0.278 + 2.05i)10-s − 4.88i·11-s + 6.31i·13-s + (3.22 − 1.90i)14-s + (3.43 + 2.05i)16-s − 4.18i·17-s + 0.299·19-s + (−0.779 + 2.82i)20-s + (0.927 − 6.84i)22-s + 4.88i·23-s + ⋯
L(s)  = 1  + (0.990 + 0.134i)2-s + (0.963 + 0.266i)4-s + 0.654i·5-s + (0.785 − 0.619i)7-s + (0.919 + 0.393i)8-s + (−0.0879 + 0.648i)10-s − 1.47i·11-s + 1.75i·13-s + (0.861 − 0.508i)14-s + (0.858 + 0.513i)16-s − 1.01i·17-s + 0.0687·19-s + (−0.174 + 0.631i)20-s + (0.197 − 1.46i)22-s + 1.01i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.921 - 0.387i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.921 - 0.387i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.06485 + 0.618494i\)
\(L(\frac12)\) \(\approx\) \(3.06485 + 0.618494i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.189i)T \)
3 \( 1 \)
7 \( 1 + (-2.07 + 1.63i)T \)
good5 \( 1 - 1.46iT - 5T^{2} \)
11 \( 1 + 4.88iT - 11T^{2} \)
13 \( 1 - 6.31iT - 13T^{2} \)
17 \( 1 + 4.18iT - 17T^{2} \)
19 \( 1 - 0.299T + 19T^{2} \)
23 \( 1 - 4.88iT - 23T^{2} \)
29 \( 1 + 3.64T + 29T^{2} \)
31 \( 1 + 5.56T + 31T^{2} \)
37 \( 1 - 3.59T + 37T^{2} \)
41 \( 1 + 4.62iT - 41T^{2} \)
43 \( 1 - 5.16iT - 43T^{2} \)
47 \( 1 + 9.24T + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 - 5.40iT - 61T^{2} \)
67 \( 1 + 6.31iT - 67T^{2} \)
71 \( 1 - 6.13iT - 71T^{2} \)
73 \( 1 - 1.89iT - 73T^{2} \)
79 \( 1 + 14.5iT - 79T^{2} \)
83 \( 1 + 14.0T + 83T^{2} \)
89 \( 1 + 13.6iT - 89T^{2} \)
97 \( 1 + 5.40iT - 97T^{2} \)
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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

−10.91074413594407237923032668714, −9.618517480598726679340245599353, −8.533657389592335753801083580829, −7.44125957988621170716937526470, −6.91044895542052378951611746033, −5.92479730354499819809089989184, −4.93715353471635984767720541338, −3.94971929222198510167906707609, −3.05039478781326385391011308621, −1.66063184999029296929821447947, 1.52534517611538518383137729204, 2.62163008687184372534546505080, 4.00174935531137292735459507379, 4.99486341057011406751085266536, 5.45210261000669826880219141224, 6.60209789916627246647838856332, 7.75030625191987119270458576372, 8.334712666805818965597395847438, 9.610083529866961093537252617887, 10.49254830209333732948911234647

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