↑

Properties

Label 2-756-28.19-c1-0-46
Degree $2$
Conductor $756$
Sign $-0.649 + 0.759i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.31 − 0.518i)2-s + (1.46 + 1.36i)4-s + (−2.03 − 1.17i)5-s + (2.03 − 1.69i)7-s + (−1.21 − 2.55i)8-s + (2.06 + 2.59i)10-s + (2.18 − 1.26i)11-s − 1.48i·13-s + (−3.55 + 1.17i)14-s + (0.274 + 3.99i)16-s + (−1.66 + 0.958i)17-s + (−0.454 + 0.786i)19-s + (−1.36 − 4.48i)20-s + (−3.53 + 0.526i)22-s + (4.55 + 2.63i)23-s + ⋯
L(s)  = 1  + (−0.930 − 0.366i)2-s + (0.730 + 0.682i)4-s + (−0.908 − 0.524i)5-s + (0.768 − 0.639i)7-s + (−0.429 − 0.902i)8-s + (0.652 + 0.821i)10-s + (0.659 − 0.380i)11-s − 0.411i·13-s + (−0.949 + 0.312i)14-s + (0.0687 + 0.997i)16-s + (−0.402 + 0.232i)17-s + (−0.104 + 0.180i)19-s + (−0.306 − 1.00i)20-s + (−0.753 + 0.112i)22-s + (0.950 + 0.548i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 + 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.649 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.649 + 0.759i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.649 + 0.759i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.296704 - 0.644177i\)
\(L(\frac12)\) \(\approx\) \(0.296704 - 0.644177i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 + 0.518i)T \)
3 \( 1 \)
7 \( 1 + (-2.03 + 1.69i)T \)
good5 \( 1 + (2.03 + 1.17i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.18 + 1.26i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.48iT - 13T^{2} \)
17 \( 1 + (1.66 - 0.958i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.454 - 0.786i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.55 - 2.63i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.85T + 29T^{2} \)
31 \( 1 + (3.77 + 6.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.63 - 8.03i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 12.0iT - 41T^{2} \)
43 \( 1 + 10.9iT - 43T^{2} \)
47 \( 1 + (-2.04 + 3.54i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.83 - 4.91i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.98 + 6.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.72 - 3.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.36 - 0.790i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.670iT - 71T^{2} \)
73 \( 1 + (-2.49 + 1.44i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (13.5 + 7.81i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.82T + 83T^{2} \)
89 \( 1 + (3.23 + 1.86i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 14.0iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.11254251627609853372077231720, −8.930055839030799339227860648278, −8.503025801074346330873741565496, −7.56308121567812642256675705427, −7.02010906228722311037995850217, −5.58081598115657218860022426909, −4.20002334265910323133619409606, −3.52879499576824052928169688984, −1.80943612739834887891082692207, −0.51043021561988712690985655496, 1.56816179653793716059637144747, 2.87923624110720072866282138296, 4.36057418234605586141712817257, 5.43501316772555385396585903937, 6.65871001689798703327341868782, 7.23959989738717346798626926526, 8.106580130658737761897775499805, 8.921041112174183625449344234033, 9.529837508329651385423344648964, 10.85400404061815868662897892190

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