Properties

Label 2-1792-8.5-c1-0-20
Degree $2$
Conductor $1792$
Sign $0.707 + 0.707i$
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  − 2.73i·3-s + 0.732i·5-s + 7-s − 4.46·9-s + 5.46i·11-s − 0.732i·13-s + 2·15-s + 0.535·17-s + 4.19i·19-s − 2.73i·21-s + 4.92·23-s + 4.46·25-s + 3.99i·27-s − 3.46i·29-s + 9.46·31-s + ⋯
L(s)  = 1  − 1.57i·3-s + 0.327i·5-s + 0.377·7-s − 1.48·9-s + 1.64i·11-s − 0.203i·13-s + 0.516·15-s + 0.129·17-s + 0.962i·19-s − 0.596i·21-s + 1.02·23-s + 0.892·25-s + 0.769i·27-s − 0.643i·29-s + 1.69·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(14.3091\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.828539103\)
\(L(\frac12)\) \(\approx\) \(1.828539103\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good3 \( 1 + 2.73iT - 3T^{2} \)
5 \( 1 - 0.732iT - 5T^{2} \)
11 \( 1 - 5.46iT - 11T^{2} \)
13 \( 1 + 0.732iT - 13T^{2} \)
17 \( 1 - 0.535T + 17T^{2} \)
19 \( 1 - 4.19iT - 19T^{2} \)
23 \( 1 - 4.92T + 23T^{2} \)
29 \( 1 + 3.46iT - 29T^{2} \)
31 \( 1 - 9.46T + 31T^{2} \)
37 \( 1 + 11.4iT - 37T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 - 9.46iT - 43T^{2} \)
47 \( 1 - 5.46T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 4.19iT - 59T^{2} \)
61 \( 1 - 13.1iT - 61T^{2} \)
67 \( 1 + 1.07iT - 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 - 4.92T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 10.7iT - 83T^{2} \)
89 \( 1 - 8.92T + 89T^{2} \)
97 \( 1 - 7.46T + 97T^{2} \)
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

−9.036873106622873092506620995167, −8.053858181209434978460107615991, −7.56043625517431947587895587913, −6.90685299064400065435569443652, −6.26411950007931699428408471275, −5.23744675204222577953929710174, −4.25348205614624460703389397183, −2.81386405086547800903504012379, −2.02091744065489837613140076739, −1.03469505783266015755837440082, 0.899703280123874887803585800784, 2.88062210475824555636089253444, 3.45451295076912313358052218485, 4.67628803967934502731106463455, 4.96319786421136577335121990107, 5.94336769717905165787924854229, 6.89985728286161736770429246732, 8.219540337327791830991409424322, 8.813360184318377885480127935647, 9.183739273255392723639773872033

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