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Properties

Label 2-756-28.27-c1-0-39
Degree $2$
Conductor $756$
Sign $0.999 - 0.0144i$
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  + (−0.748 + 1.19i)2-s + (−0.879 − 1.79i)4-s + 3.90i·5-s + (1.12 − 2.39i)7-s + (2.81 + 0.288i)8-s + (−4.69 − 2.92i)10-s − 5.01i·11-s − 5.59i·13-s + (2.02 + 3.14i)14-s + (−2.45 + 3.15i)16-s − 2.41i·17-s + 3.11·19-s + (7.02 − 3.43i)20-s + (6.02 + 3.75i)22-s − 3.45i·23-s + ⋯
L(s)  = 1  + (−0.529 + 0.848i)2-s + (−0.439 − 0.898i)4-s + 1.74i·5-s + (0.426 − 0.904i)7-s + (0.994 + 0.102i)8-s + (−1.48 − 0.925i)10-s − 1.51i·11-s − 1.55i·13-s + (0.541 + 0.840i)14-s + (−0.613 + 0.789i)16-s − 0.584i·17-s + 0.714·19-s + (1.57 − 0.769i)20-s + (1.28 + 0.800i)22-s − 0.719i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0144i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.999 - 0.0144i$
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: \((2,\ 756,\ (\ :1/2),\ 0.999 - 0.0144i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04516 + 0.00754498i\)
\(L(\frac12)\) \(\approx\) \(1.04516 + 0.00754498i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.748 - 1.19i)T \)
3 \( 1 \)
7 \( 1 + (-1.12 + 2.39i)T \)
good5 \( 1 - 3.90iT - 5T^{2} \)
11 \( 1 + 5.01iT - 11T^{2} \)
13 \( 1 + 5.59iT - 13T^{2} \)
17 \( 1 + 2.41iT - 17T^{2} \)
19 \( 1 - 3.11T + 19T^{2} \)
23 \( 1 + 3.45iT - 23T^{2} \)
29 \( 1 + 4.76T + 29T^{2} \)
31 \( 1 - 0.482T + 31T^{2} \)
37 \( 1 - 5.90T + 37T^{2} \)
41 \( 1 - 1.51iT - 41T^{2} \)
43 \( 1 - 1.77iT - 43T^{2} \)
47 \( 1 - 2.38T + 47T^{2} \)
53 \( 1 - 5.62T + 53T^{2} \)
59 \( 1 - 6.38T + 59T^{2} \)
61 \( 1 + 6.94iT - 61T^{2} \)
67 \( 1 + 3.92iT - 67T^{2} \)
71 \( 1 - 8.25iT - 71T^{2} \)
73 \( 1 - 3.59iT - 73T^{2} \)
79 \( 1 + 9.32iT - 79T^{2} \)
83 \( 1 + 5.76T + 83T^{2} \)
89 \( 1 - 17.6iT - 89T^{2} \)
97 \( 1 - 15.5iT - 97T^{2} \)
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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

−10.40513158799828508983357898675, −9.615319355414981016271665937027, −8.275591452097569299694879553142, −7.70384398199529695620862244539, −6.99035067340687842857444188683, −6.11688182176564826390350630423, −5.33169419723507549563772542947, −3.78074022765668739385707115966, −2.79016891185701366563285309723, −0.69290676242176204297093508278, 1.46540096697595582490806801792, 2.12398225480806904961248287422, 4.05244803419222570894961113461, 4.66594296702018665805910072345, 5.57074921477816044365785215274, 7.21929053766885457151115520546, 8.066296433275583278098522760182, 9.055657721908317787267212079283, 9.234346760367880312630599017107, 10.07189680434514837216946514230

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