Properties

Degree $2$
Conductor $1764$
Sign $0.966 + 0.257i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (1.05 − 0.946i)2-s + (0.207 − 1.98i)4-s + 1.60i·5-s + (−1.66 − 2.28i)8-s + (1.52 + 1.68i)10-s + 2.67i·11-s + 3.37i·13-s + (−3.91 − 0.823i)16-s + 1.60i·17-s + 4.30·19-s + (3.19 + 0.333i)20-s + (2.53 + 2.81i)22-s − 6.46i·23-s + 2.41·25-s + (3.19 + 3.54i)26-s + ⋯
L(s)  = 1  + (0.742 − 0.669i)2-s + (0.103 − 0.994i)4-s + 0.719i·5-s + (−0.588 − 0.808i)8-s + (0.481 + 0.534i)10-s + 0.807i·11-s + 0.937i·13-s + (−0.978 − 0.205i)16-s + 0.390i·17-s + 0.987·19-s + (0.715 + 0.0744i)20-s + (0.540 + 0.599i)22-s − 1.34i·23-s + 0.482·25-s + (0.627 + 0.696i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.966 + 0.257i$
Motivic weight: \(1\)
Character: $\chi_{1764} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.966 + 0.257i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.618050894\)
\(L(\frac12)\) \(\approx\) \(2.618050894\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.05 + 0.946i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 1.60iT - 5T^{2} \)
11 \( 1 - 2.67iT - 11T^{2} \)
13 \( 1 - 3.37iT - 13T^{2} \)
17 \( 1 - 1.60iT - 17T^{2} \)
19 \( 1 - 4.30T + 19T^{2} \)
23 \( 1 + 6.46iT - 23T^{2} \)
29 \( 1 - 4.71T + 29T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
37 \( 1 - 0.242T + 37T^{2} \)
41 \( 1 - 11.6iT - 41T^{2} \)
43 \( 1 - 7.95iT - 43T^{2} \)
47 \( 1 - 9.04T + 47T^{2} \)
53 \( 1 + 2.46T + 53T^{2} \)
59 \( 1 + 9.04T + 59T^{2} \)
61 \( 1 + 3.37iT - 61T^{2} \)
67 \( 1 + 11.2iT - 67T^{2} \)
71 \( 1 - 8.03iT - 71T^{2} \)
73 \( 1 + 6.17iT - 73T^{2} \)
79 \( 1 + 3.29iT - 79T^{2} \)
83 \( 1 + 12.7T + 83T^{2} \)
89 \( 1 - 0.275iT - 89T^{2} \)
97 \( 1 + 12.9iT - 97T^{2} \)
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

−9.624080018888830204457890779492, −8.606744899494740712107482996757, −7.49864313832611087491179166168, −6.52300074357902980316541416523, −6.25162813535621545611958574358, −4.74551640178249071325326318152, −4.46452844879306116732147435930, −3.13971673449761748688163637606, −2.49452483792109241774790793366, −1.24661757950307111564011727739, 0.895560000748961543923742238190, 2.72123074924559231606282773454, 3.49461750283525887399235849346, 4.54713328602788324080118803283, 5.41706422315206395137461283797, 5.80382794282878885860936656525, 6.96896808986232599805114159858, 7.66986158437648202269050204641, 8.469335052098607995126435349539, 8.994523011124562910248395914978

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