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Properties

Label 2-756-36.11-c1-0-12
Degree $2$
Conductor $756$
Sign $0.911 - 0.410i$
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  + (−1.37 + 0.318i)2-s + (1.79 − 0.878i)4-s + (−2.51 + 1.45i)5-s + (−0.866 − 0.5i)7-s + (−2.19 + 1.78i)8-s + (3.00 − 2.80i)10-s + (1.18 − 2.05i)11-s + (−0.125 − 0.218i)13-s + (1.35 + 0.412i)14-s + (2.45 − 3.15i)16-s − 7.60i·17-s + 2.60i·19-s + (−3.24 + 4.82i)20-s + (−0.978 + 3.20i)22-s + (4.51 + 7.82i)23-s + ⋯
L(s)  = 1  + (−0.974 + 0.225i)2-s + (0.898 − 0.439i)4-s + (−1.12 + 0.650i)5-s + (−0.327 − 0.188i)7-s + (−0.776 + 0.630i)8-s + (0.950 − 0.887i)10-s + (0.357 − 0.618i)11-s + (−0.0349 − 0.0604i)13-s + (0.361 + 0.110i)14-s + (0.614 − 0.788i)16-s − 1.84i·17-s + 0.598i·19-s + (−0.726 + 1.07i)20-s + (−0.208 + 0.683i)22-s + (0.941 + 1.63i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.911 - 0.410i$
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: \((2,\ 756,\ (\ :1/2),\ 0.911 - 0.410i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.711989 + 0.153027i\)
\(L(\frac12)\) \(\approx\) \(0.711989 + 0.153027i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.318i)T \)
3 \( 1 \)
7 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (2.51 - 1.45i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.18 + 2.05i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.125 + 0.218i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 7.60iT - 17T^{2} \)
19 \( 1 - 2.60iT - 19T^{2} \)
23 \( 1 + (-4.51 - 7.82i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.53 - 3.19i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.0905 + 0.0522i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.15T + 37T^{2} \)
41 \( 1 + (-7.24 + 4.18i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.18 - 4.14i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.248 + 0.430i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.38iT - 53T^{2} \)
59 \( 1 + (-3.99 - 6.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.69 + 6.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.444 - 0.256i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.45T + 71T^{2} \)
73 \( 1 - 4.06T + 73T^{2} \)
79 \( 1 + (-10.0 - 5.79i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.439 - 0.761i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 14.6iT - 89T^{2} \)
97 \( 1 + (-2.88 + 4.98i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.37630522514511390980872635634, −9.430877616429807383037979491819, −8.760730535536489880580345595809, −7.52684341044252982404324670151, −7.35060129103146791013360235655, −6.32812498462899369738106976610, −5.17597036850546243834944745423, −3.60875473313854681270487091936, −2.80887550243173225489555868865, −0.840978450541294228108748312321, 0.788935454230874081292269447772, 2.37478294792159185869580899817, 3.74375074349656472741430383852, 4.59659082855546254128275061047, 6.20562473179535365397976047705, 6.97309235315876721187242451669, 8.013239756396151641116136868511, 8.554311467499565264640504007003, 9.265568871560372489057485017735, 10.32652463333702799660902337815

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