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Properties

Label 2-756-36.23-c1-0-13
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: $\chi_{756} (71, \cdot )$
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

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

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