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Properties

Label 2-756-189.5-c1-0-14
Degree $2$
Conductor $756$
Sign $0.944 - 0.329i$
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  + (0.212 + 1.71i)3-s + (0.455 − 2.58i)5-s + (2.64 − 0.0947i)7-s + (−2.90 + 0.730i)9-s + (0.625 − 0.110i)11-s + (−2.83 + 3.37i)13-s + (4.53 + 0.234i)15-s + 8.05·17-s − 3.96i·19-s + (0.724 + 4.52i)21-s + (3.59 − 4.28i)23-s + (−1.77 − 0.645i)25-s + (−1.87 − 4.84i)27-s + (4.88 + 5.82i)29-s + (1.20 + 3.30i)31-s + ⋯
L(s)  = 1  + (0.122 + 0.992i)3-s + (0.203 − 1.15i)5-s + (0.999 − 0.0358i)7-s + (−0.969 + 0.243i)9-s + (0.188 − 0.0332i)11-s + (−0.785 + 0.935i)13-s + (1.17 + 0.0604i)15-s + 1.95·17-s − 0.909i·19-s + (0.158 + 0.987i)21-s + (0.749 − 0.893i)23-s + (−0.354 − 0.129i)25-s + (−0.360 − 0.932i)27-s + (0.907 + 1.08i)29-s + (0.215 + 0.592i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.944 - 0.329i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.944 - 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81480 + 0.308016i\)
\(L(\frac12)\) \(\approx\) \(1.81480 + 0.308016i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.212 - 1.71i)T \)
7 \( 1 + (-2.64 + 0.0947i)T \)
good5 \( 1 + (-0.455 + 2.58i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-0.625 + 0.110i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (2.83 - 3.37i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 - 8.05T + 17T^{2} \)
19 \( 1 + 3.96iT - 19T^{2} \)
23 \( 1 + (-3.59 + 4.28i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-4.88 - 5.82i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.20 - 3.30i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (3.51 + 6.08i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.16 - 3.49i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (0.637 + 0.231i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (2.65 + 0.964i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (3.98 - 2.29i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.846 + 0.710i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.698 - 1.91i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (2.32 - 13.1i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-6.01 - 3.47i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.04 - 2.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.57 + 14.6i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.855 + 0.717i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + 13.9T + 89T^{2} \)
97 \( 1 + (5.47 - 15.0i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.31424849624349097832037201536, −9.403007420134238594623995731803, −8.826839877421407051361674377603, −8.124478622334661258887859701426, −6.98377351399096783173028766408, −5.44816715793277049121692197937, −4.94811326984892858367121727806, −4.26371697657650385443560144913, −2.83327940988188864213882323584, −1.24029554279515543923835707835, 1.27633577433189017356377953935, 2.57456411454410439680843286998, 3.42576900299347521262055920174, 5.17846364382042531489232096526, 5.93787593792106561997322519745, 6.94129091600146274288557273914, 7.913684337773337790046130103571, 7.988163536747786121367153943995, 9.595027969266464865454872829631, 10.33312262272769602569735823750

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