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Properties

Label 2-756-252.31-c1-0-28
Degree $2$
Conductor $756$
Sign $0.110 + 0.993i$
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  + (−1.36 + 0.366i)2-s + (1.73 − i)4-s − 3.46i·5-s + (1.73 − 2i)7-s + (−1.99 + 2i)8-s + (1.26 + 4.73i)10-s − 2i·11-s + (4.5 + 2.59i)13-s + (−1.63 + 3.36i)14-s + (1.99 − 3.46i)16-s + (4.5 + 2.59i)17-s + (−0.866 − 1.5i)19-s + (−3.46 − 5.99i)20-s + (0.732 + 2.73i)22-s − 4i·23-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.5i)4-s − 1.54i·5-s + (0.654 − 0.755i)7-s + (−0.707 + 0.707i)8-s + (0.400 + 1.49i)10-s − 0.603i·11-s + (1.24 + 0.720i)13-s + (−0.436 + 0.899i)14-s + (0.499 − 0.866i)16-s + (1.09 + 0.630i)17-s + (−0.198 − 0.344i)19-s + (−0.774 − 1.34i)20-s + (0.156 + 0.582i)22-s − 0.834i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.110 + 0.993i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.110 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.816717 - 0.730629i\)
\(L(\frac12)\) \(\approx\) \(0.816717 - 0.730629i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
3 \( 1 \)
7 \( 1 + (-1.73 + 2i)T \)
good5 \( 1 + 3.46iT - 5T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + (-4.5 - 2.59i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.5 - 2.59i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.866 + 1.5i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.59 + 4.5i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 + 0.866i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (9.52 - 5.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.866 + 1.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.866 + 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 - 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.79 + 4.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2iT - 71T^{2} \)
73 \( 1 + (10.5 + 6.06i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.59 - 1.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.59 - 4.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.5 - 4.33i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.5 + 0.866i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.04210042682366934076245216455, −9.017888186806518259137530082489, −8.454510595502661222540850457838, −7.969221818795978020139748941146, −6.75686167814391040256451149631, −5.77855658012803841327457622855, −4.84598698349152201866765715850, −3.69670605089830530733905766987, −1.66568231060672262106481821318, −0.851061924470404045000341919260, 1.62510813136776771029520655091, 2.82794176376083028553456378375, 3.58315216467576070861443692640, 5.46872267613228869089793733454, 6.36831390430986484742052451537, 7.27487224135454540007348829928, 7.985746854114892876952248189650, 8.788552044798577020095608710502, 9.987814886605813661477321959985, 10.34012909309655415323109956824

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