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Properties

Label 2-756-84.11-c1-0-28
Degree $2$
Conductor $756$
Sign $0.999 + 0.00601i$
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 − 0.0220i)2-s + (1.99 + 0.0624i)4-s + (0.303 − 0.175i)5-s + (2.63 − 0.234i)7-s + (−2.82 − 0.132i)8-s + (−0.432 + 0.240i)10-s + (−0.356 + 0.617i)11-s + 0.127·13-s + (−3.73 + 0.272i)14-s + (3.99 + 0.249i)16-s + (5.30 + 3.06i)17-s + (2.91 − 1.68i)19-s + (0.617 − 0.331i)20-s + (0.517 − 0.864i)22-s + (−2.38 − 4.13i)23-s + ⋯
L(s)  = 1  + (−0.999 − 0.0156i)2-s + (0.999 + 0.0312i)4-s + (0.135 − 0.0783i)5-s + (0.996 − 0.0884i)7-s + (−0.998 − 0.0467i)8-s + (−0.136 + 0.0761i)10-s + (−0.107 + 0.186i)11-s + 0.0353·13-s + (−0.997 + 0.0729i)14-s + (0.998 + 0.0623i)16-s + (1.28 + 0.742i)17-s + (0.669 − 0.386i)19-s + (0.138 − 0.0740i)20-s + (0.110 − 0.184i)22-s + (−0.497 − 0.862i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.999 + 0.00601i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.999 + 0.00601i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16845 - 0.00351205i\)
\(L(\frac12)\) \(\approx\) \(1.16845 - 0.00351205i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0220i)T \)
3 \( 1 \)
7 \( 1 + (-2.63 + 0.234i)T \)
good5 \( 1 + (-0.303 + 0.175i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.356 - 0.617i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.127T + 13T^{2} \)
17 \( 1 + (-5.30 - 3.06i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.91 + 1.68i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.38 + 4.13i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.39iT - 29T^{2} \)
31 \( 1 + (-0.00202 - 0.00116i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.17 - 3.76i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.22iT - 41T^{2} \)
43 \( 1 + 7.55iT - 43T^{2} \)
47 \( 1 + (1.33 + 2.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.1 - 5.86i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.13 + 8.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.83 + 4.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.79 - 4.50i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.61T + 71T^{2} \)
73 \( 1 + (2.15 - 3.72i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.92 - 1.10i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.12T + 83T^{2} \)
89 \( 1 + (-7.79 + 4.49i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
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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

−10.17906228139362447042367797177, −9.611270829843934080384672426882, −8.474176469403027071794591686589, −7.960526699568879895213724278223, −7.15549825501456414003125762262, −6.01898744213594712729100678861, −5.12429233589498662945089389270, −3.69376710042693358192151142896, −2.29211707245309797855285307946, −1.13307613827685069663468974171, 1.10620450757498835715946022445, 2.34970619605160629757955627748, 3.63241190182553541966401473878, 5.23958559427276440657884633086, 5.92649768301066402991295850872, 7.26458056783640086984692024774, 7.79221604971070266375872999244, 8.587491193396474060034094977044, 9.560349781123567382786463131383, 10.19223186519523548094399833424

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