Properties

Label 2-756-36.11-c1-0-17
Degree $2$
Conductor $756$
Sign $0.985 + 0.167i$
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  + (−0.908 + 1.08i)2-s + (−0.348 − 1.96i)4-s + (−2.24 + 1.29i)5-s + (−0.866 − 0.5i)7-s + (2.45 + 1.41i)8-s + (0.635 − 3.60i)10-s + (0.124 − 0.216i)11-s + (−0.0646 − 0.111i)13-s + (1.32 − 0.484i)14-s + (−3.75 + 1.37i)16-s + 0.554i·17-s − 3.58i·19-s + (3.33 + 3.96i)20-s + (0.120 + 0.331i)22-s + (−3.94 − 6.83i)23-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)2-s + (−0.174 − 0.984i)4-s + (−1.00 + 0.579i)5-s + (−0.327 − 0.188i)7-s + (0.866 + 0.499i)8-s + (0.200 − 1.14i)10-s + (0.0376 − 0.0651i)11-s + (−0.0179 − 0.0310i)13-s + (0.355 − 0.129i)14-s + (−0.939 + 0.342i)16-s + 0.134i·17-s − 0.823i·19-s + (0.744 + 0.886i)20-s + (0.0257 + 0.0707i)22-s + (−0.823 − 1.42i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.985 + 0.167i$
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: \((2,\ 756,\ (\ :1/2),\ 0.985 + 0.167i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.682309 - 0.0576456i\)
\(L(\frac12)\) \(\approx\) \(0.682309 - 0.0576456i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.908 - 1.08i)T \)
3 \( 1 \)
7 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (2.24 - 1.29i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.124 + 0.216i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.0646 + 0.111i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.554iT - 17T^{2} \)
19 \( 1 + 3.58iT - 19T^{2} \)
23 \( 1 + (3.94 + 6.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.90 - 3.40i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.96 + 4.02i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.96T + 37T^{2} \)
41 \( 1 + (4.25 - 2.45i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.78 - 2.18i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.41 + 4.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.00iT - 53T^{2} \)
59 \( 1 + (3.71 + 6.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.42 + 11.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.23 + 3.02i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.7T + 71T^{2} \)
73 \( 1 + 14.1T + 73T^{2} \)
79 \( 1 + (-2.04 - 1.17i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.36 + 4.09i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.82iT - 89T^{2} \)
97 \( 1 + (5.67 - 9.82i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.22279318227004303191430284722, −9.439675147022665405990163107299, −8.282296693344070313136357676164, −7.908637010409150145530152432785, −6.73859871020144156564682373107, −6.40508491936550558733566626772, −4.94213856597505099802203471121, −4.01284786875429801784550929967, −2.59284378311707346642825685806, −0.54363990453613199050546736402, 1.07170220648840418017056846471, 2.66168234948920160391965753726, 3.82500804469334230967727917293, 4.52562572112666222344376762261, 5.96975194097338020180855911637, 7.28886878738043883783618367039, 8.021781489030598832957405980464, 8.629491716934420126097259593969, 9.624403855769584073945876619610, 10.23547216849711308989284601561

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