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Properties

Label 2-756-36.11-c1-0-20
Degree $2$
Conductor $756$
Sign $0.730 - 0.682i$
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  + (0.0405 + 1.41i)2-s + (−1.99 + 0.114i)4-s + (2.16 − 1.25i)5-s + (0.866 + 0.5i)7-s + (−0.243 − 2.81i)8-s + (1.85 + 3.01i)10-s + (0.351 − 0.608i)11-s + (1.55 + 2.69i)13-s + (−0.671 + 1.24i)14-s + (3.97 − 0.458i)16-s − 7.91i·17-s − 2.37i·19-s + (−4.18 + 2.74i)20-s + (0.874 + 0.472i)22-s + (0.346 + 0.600i)23-s + ⋯
L(s)  = 1  + (0.0287 + 0.999i)2-s + (−0.998 + 0.0573i)4-s + (0.969 − 0.559i)5-s + (0.327 + 0.188i)7-s + (−0.0860 − 0.996i)8-s + (0.587 + 0.953i)10-s + (0.105 − 0.183i)11-s + (0.430 + 0.746i)13-s + (−0.179 + 0.332i)14-s + (0.993 − 0.114i)16-s − 1.91i·17-s − 0.544i·19-s + (−0.935 + 0.614i)20-s + (0.186 + 0.100i)22-s + (0.0722 + 0.125i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.730 - 0.682i$
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: \((2,\ 756,\ (\ :1/2),\ 0.730 - 0.682i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63096 + 0.643643i\)
\(L(\frac12)\) \(\approx\) \(1.63096 + 0.643643i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0405 - 1.41i)T \)
3 \( 1 \)
7 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + (-2.16 + 1.25i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.351 + 0.608i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.55 - 2.69i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 7.91iT - 17T^{2} \)
19 \( 1 + 2.37iT - 19T^{2} \)
23 \( 1 + (-0.346 - 0.600i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.70 - 5.02i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.34 + 4.24i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.85T + 37T^{2} \)
41 \( 1 + (5.82 - 3.36i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.17 - 1.83i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.67 - 6.37i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.889iT - 53T^{2} \)
59 \( 1 + (3.63 + 6.28i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.39 - 4.15i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.40 - 4.27i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.34T + 71T^{2} \)
73 \( 1 - 8.20T + 73T^{2} \)
79 \( 1 + (-2.27 - 1.31i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.50 + 11.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 1.71iT - 89T^{2} \)
97 \( 1 + (6.81 - 11.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.04767779958845649175488964889, −9.322282423153374825525188122450, −8.838281519386772814599076890045, −7.85985924931647150432039883276, −6.79707884175928623966907769202, −6.11993807793031025989086228678, −5.04348875483966466903637102962, −4.57614059684848263943241491247, −2.87325795583449039032725825188, −1.12763042956964382613694842909, 1.34678063505792982265779236921, 2.42480217734589929934192339488, 3.54885326153947941349129803446, 4.60851943089103208035161875693, 5.79581857414976926784660229180, 6.45055651919538800729800614359, 8.059988067481650434147757028527, 8.542914377674453767253643913579, 9.802586552487525397735019778169, 10.42006663248369211376478282146

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