↑

Properties

Label 2-756-21.2-c2-0-9
Degree $2$
Conductor $756$
Sign $0.832 - 0.553i$
Analytic cond. $20.5995$
Root an. cond. $4.53866$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (7.74 + 4.47i)5-s − 7·7-s + (7.74 − 4.47i)11-s + 3·13-s + (7.74 − 4.47i)17-s + (11 − 19.0i)19-s + (15.4 + 8.94i)23-s + (27.5 + 47.6i)25-s − 8.94i·29-s + (16.5 + 28.5i)31-s + (−54.2 − 31.3i)35-s + (−5.5 + 9.52i)37-s + 80.4i·41-s + 59·43-s + (−69.7 − 40.2i)47-s + ⋯
L(s)  = 1  + (1.54 + 0.894i)5-s − 7-s + (0.704 − 0.406i)11-s + 0.230·13-s + (0.455 − 0.263i)17-s + (0.578 − 1.00i)19-s + (0.673 + 0.388i)23-s + (1.10 + 1.90i)25-s − 0.308i·29-s + (0.532 + 0.921i)31-s + (−1.54 − 0.894i)35-s + (−0.148 + 0.257i)37-s + 1.96i·41-s + 1.37·43-s + (−1.48 − 0.856i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(20.5995\)
Root analytic conductor: \(4.53866\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1),\ 0.832 - 0.553i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.412910763\)
\(L(\frac12)\) \(\approx\) \(2.412910763\)
\(L(2)\) 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 \( 1 + 7T \)
good5 \( 1 + (-7.74 - 4.47i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-7.74 + 4.47i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 3T + 169T^{2} \)
17 \( 1 + (-7.74 + 4.47i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-11 + 19.0i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-15.4 - 8.94i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 8.94iT - 841T^{2} \)
31 \( 1 + (-16.5 - 28.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (5.5 - 9.52i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 80.4iT - 1.68e3T^{2} \)
43 \( 1 - 59T + 1.84e3T^{2} \)
47 \( 1 + (69.7 + 40.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (7.74 - 4.47i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (49.5 - 85.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 98.3iT - 5.04e3T^{2} \)
73 \( 1 + (-31 - 53.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (31.5 - 54.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 89.4iT - 6.88e3T^{2} \)
89 \( 1 + (-23.2 - 13.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 127T + 9.40e3T^{2} \)
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

−10.04967033965484572870124244594, −9.513925862105959646758423598308, −8.860643565091056203958489195900, −7.33697922375243622170157822605, −6.52085806051182568893149381884, −6.06716717630518438956020274639, −5.00469146932523129916336074043, −3.32802452633330382584480182995, −2.73215060262311658242118196985, −1.23178464574414495672849603302, 1.00681899467459603939068923012, 2.12326701991976411876302400812, 3.49200632288317600556773137816, 4.73112210990119140432065238104, 5.83584508444882193150309056460, 6.22264372226963723454272584316, 7.37770638847468677710784806187, 8.681123507922296730120638230781, 9.334636030680648101263535845353, 9.869637467852446759089864941374

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