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Properties

Label 2-756-36.11-c1-0-27
Degree $2$
Conductor $756$
Sign $0.959 + 0.280i$
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  + (−0.246 + 1.39i)2-s + (−1.87 − 0.686i)4-s + (3.28 − 1.89i)5-s + (−0.866 − 0.5i)7-s + (1.41 − 2.44i)8-s + (1.83 + 5.04i)10-s + (−0.147 + 0.255i)11-s + (−1.99 − 3.45i)13-s + (0.909 − 1.08i)14-s + (3.05 + 2.58i)16-s + 3.60i·17-s − 6.02i·19-s + (−7.47 + 1.30i)20-s + (−0.318 − 0.267i)22-s + (−2.37 − 4.11i)23-s + ⋯
L(s)  = 1  + (−0.174 + 0.984i)2-s + (−0.939 − 0.343i)4-s + (1.46 − 0.848i)5-s + (−0.327 − 0.188i)7-s + (0.501 − 0.864i)8-s + (0.579 + 1.59i)10-s + (−0.0443 + 0.0768i)11-s + (−0.552 − 0.957i)13-s + (0.243 − 0.289i)14-s + (0.764 + 0.645i)16-s + 0.875i·17-s − 1.38i·19-s + (−1.67 + 0.292i)20-s + (−0.0679 − 0.0571i)22-s + (−0.495 − 0.857i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.959 + 0.280i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.959 + 0.280i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41913 - 0.203041i\)
\(L(\frac12)\) \(\approx\) \(1.41913 - 0.203041i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.246 - 1.39i)T \)
3 \( 1 \)
7 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (-3.28 + 1.89i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.147 - 0.255i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.99 + 3.45i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.60iT - 17T^{2} \)
19 \( 1 + 6.02iT - 19T^{2} \)
23 \( 1 + (2.37 + 4.11i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.904 + 0.522i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.56 + 3.21i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.86T + 37T^{2} \)
41 \( 1 + (-10.6 + 6.17i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.852 + 0.492i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.68 - 2.91i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.03iT - 53T^{2} \)
59 \( 1 + (-4.15 - 7.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.48 - 9.50i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.26 + 5.34i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.73T + 71T^{2} \)
73 \( 1 - 8.63T + 73T^{2} \)
79 \( 1 + (-4.74 - 2.74i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.07 - 7.05i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.35iT - 89T^{2} \)
97 \( 1 + (-6.74 + 11.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.06417853965049360256267067246, −9.328677656510814279491546591745, −8.670101868889204118878854622259, −7.75151172948701874848781539322, −6.63329916096039839621377048216, −5.90407188147897425967198836965, −5.17609123294854425869851581468, −4.26898342909696814681047869023, −2.44768821140051054497468468069, −0.805441970490700751798648302767, 1.68351619960121144817490043372, 2.54125996457374288947398058330, 3.54731417465753417605211070903, 4.94197285978211301198583801771, 5.89130674716045324702985589267, 6.81151789018982374991153429066, 7.935583104668669562970201137224, 9.182466661772090085902737155598, 9.733189786743440050797931427137, 10.16605192375595649434140050747

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