↑

Properties

Label 2-1792-28.27-c1-0-17
Degree $2$
Conductor $1792$
Sign $-i$
Analytic cond. $14.3091$
Root an. cond. $3.78274$
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  + 1.78·3-s − 1.23i·5-s + 2.64i·7-s + 0.171·9-s + 7.17i·13-s − 2.19i·15-s − 6.81·19-s + 4.71i·21-s + 7.48i·23-s + 3.48·25-s − 5.03·27-s + 3.25·35-s + 12.7i·39-s − 0.211i·45-s − 7.00·49-s + ⋯
L(s)  = 1  + 1.02·3-s − 0.550i·5-s + 0.999i·7-s + 0.0571·9-s + 1.98i·13-s − 0.565i·15-s − 1.56·19-s + 1.02i·21-s + 1.56i·23-s + 0.697·25-s − 0.969·27-s + 0.550·35-s + 2.04i·39-s − 0.0314i·45-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-i$
Analytic conductor: \(14.3091\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1791, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.879445672\)
\(L(\frac12)\) \(\approx\) \(1.879445672\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - 2.64iT \)
good3 \( 1 - 1.78T + 3T^{2} \)
5 \( 1 + 1.23iT - 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 7.17iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 6.81T + 19T^{2} \)
23 \( 1 - 7.48iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 13.9T + 59T^{2} \)
61 \( 1 + 10.6iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 15.8iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 5.29iT - 79T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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

−9.242619997775662622657115438043, −8.700961157737618148181763598445, −8.288843806241042576266023887625, −7.15444106106500613111511219195, −6.33760891764685004028615167481, −5.38049723094501658446370448400, −4.41981789378745718029394158879, −3.57420121635516214827238509819, −2.38282262187511735976021785188, −1.75761622353604833143890130413, 0.57532414993034553444522376973, 2.30928586644260383598414412713, 3.04783527168035286117061215094, 3.83374050397807626923635518839, 4.82744658999535426897083801693, 6.01655710119323559253070416834, 6.82133093365901705833919175519, 7.68954649922195660120341488063, 8.279370857850846343034234041250, 8.849763987275099388273839116195

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