↑

Properties

Label 2-756-21.17-c1-0-6
Degree $2$
Conductor $756$
Sign $0.832 - 0.553i$
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  + (1.56 + 2.70i)5-s + (2.37 − 1.17i)7-s + (3.23 + 1.87i)11-s − 2.93i·13-s + (2.71 − 4.70i)17-s + (−6.07 + 3.50i)19-s + (−1.44 + 0.832i)23-s + (−2.37 + 4.10i)25-s + 7.07i·29-s + (6.60 + 3.81i)31-s + (6.87 + 4.57i)35-s + (−4.03 − 6.98i)37-s + 9.60·41-s − 5.74·43-s + (1.56 + 2.70i)47-s + ⋯
L(s)  = 1  + (0.697 + 1.20i)5-s + (0.895 − 0.444i)7-s + (0.976 + 0.563i)11-s − 0.812i·13-s + (0.658 − 1.14i)17-s + (−1.39 + 0.804i)19-s + (−0.300 + 0.173i)23-s + (−0.474 + 0.821i)25-s + 1.31i·29-s + (1.18 + 0.685i)31-s + (1.16 + 0.773i)35-s + (−0.663 − 1.14i)37-s + 1.49·41-s − 0.875·43-s + (0.227 + 0.394i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.832 - 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88416 + 0.569422i\)
\(L(\frac12)\) \(\approx\) \(1.88416 + 0.569422i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-2.37 + 1.17i)T \)
good5 \( 1 + (-1.56 - 2.70i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.23 - 1.87i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.93iT - 13T^{2} \)
17 \( 1 + (-2.71 + 4.70i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.07 - 3.50i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.44 - 0.832i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.07iT - 29T^{2} \)
31 \( 1 + (-6.60 - 3.81i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.03 + 6.98i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.60T + 41T^{2} \)
43 \( 1 + 5.74T + 43T^{2} \)
47 \( 1 + (-1.56 - 2.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.11 + 1.79i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.20 - 3.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.57 - 1.48i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.90 + 8.50i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.33iT - 71T^{2} \)
73 \( 1 + (-2.46 - 1.42i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.57 + 4.45i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.6T + 83T^{2} \)
89 \( 1 + (-2.59 - 4.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.39iT - 97T^{2} \)
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

−10.44816494764851221685249636768, −9.809871061766522927713542673405, −8.728384231470793360602856602063, −7.67081140994425536217531462215, −6.95739215736572525994737269857, −6.12582561143418168427643535094, −5.06746695494155694985718881715, −3.91067999141104340574854238629, −2.73507863402643284123443737634, −1.52411431677430727389562670707, 1.23827794568111362717939462634, 2.21158311085030236538801099781, 4.10242690499596127459282104278, 4.74428435127236175485680379300, 5.89656483838328032582453526350, 6.45056854162401002763317952973, 8.066411738866566524789583788741, 8.590636162437739416950665444068, 9.234633130988771557749930673605, 10.13495650201694718808343969588

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