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Properties

Label 2-756-84.11-c1-0-43
Degree $2$
Conductor $756$
Sign $0.655 + 0.755i$
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  + (−1.38 + 0.290i)2-s + (1.83 − 0.803i)4-s + (2.75 − 1.59i)5-s + (−0.944 − 2.47i)7-s + (−2.30 + 1.64i)8-s + (−3.35 + 3.00i)10-s + (−1.24 + 2.16i)11-s + 6.61·13-s + (2.02 + 3.14i)14-s + (2.70 − 2.94i)16-s + (−2.67 − 1.54i)17-s + (4.40 − 2.54i)19-s + (3.77 − 5.13i)20-s + (1.10 − 3.35i)22-s + (2.53 + 4.39i)23-s + ⋯
L(s)  = 1  + (−0.978 + 0.205i)2-s + (0.915 − 0.401i)4-s + (1.23 − 0.712i)5-s + (−0.356 − 0.934i)7-s + (−0.813 + 0.581i)8-s + (−1.06 + 0.950i)10-s + (−0.376 + 0.651i)11-s + 1.83·13-s + (0.541 + 0.841i)14-s + (0.677 − 0.735i)16-s + (−0.649 − 0.374i)17-s + (1.01 − 0.583i)19-s + (0.843 − 1.14i)20-s + (0.234 − 0.715i)22-s + (0.529 + 0.916i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.655 + 0.755i$
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: \((2,\ 756,\ (\ :1/2),\ 0.655 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12023 - 0.510910i\)
\(L(\frac12)\) \(\approx\) \(1.12023 - 0.510910i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.290i)T \)
3 \( 1 \)
7 \( 1 + (0.944 + 2.47i)T \)
good5 \( 1 + (-2.75 + 1.59i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.24 - 2.16i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.61T + 13T^{2} \)
17 \( 1 + (2.67 + 1.54i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.40 + 2.54i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.53 - 4.39i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.59iT - 29T^{2} \)
31 \( 1 + (7.31 + 4.22i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.357 + 0.619i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 12.2iT - 41T^{2} \)
43 \( 1 - 3.72iT - 43T^{2} \)
47 \( 1 + (2.51 + 4.35i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.21 - 4.16i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.36 + 9.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.997 - 1.72i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.00277 + 0.00160i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + (-0.957 + 1.65i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.11 + 3.52i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.93T + 83T^{2} \)
89 \( 1 + (-0.482 + 0.278i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.127T + 97T^{2} \)
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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

−10.07195661638257625277269277407, −9.230100956270440955161134748002, −8.894369079796567653796300946561, −7.59495451267344691565799339318, −6.89303436090117390677666971047, −5.90440627030908151098324707479, −5.14247673414405268284249176081, −3.56001773462094044052651420844, −2.00065862674740117335642566580, −0.951578378545442444932571706260, 1.48633066177397697851776864827, 2.64289517178814793044618711399, 3.45709807549587608663047495217, 5.62578283613791595476424119198, 6.14263812303638693839141812276, 6.84810611377577552252353384630, 8.241777604977110366587061699626, 8.831072676258765914037782453930, 9.569553122435831387628409966072, 10.46202240987714404631725597158

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