Properties

Label 2-1792-448.237-c0-0-0
Degree $2$
Conductor $1792$
Sign $0.956 - 0.290i$
Analytic cond. $0.894324$
Root an. cond. $0.945687$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.382 − 0.923i)7-s + (−0.382 + 0.923i)9-s + (1.63 + 1.08i)11-s + (0.707 + 0.292i)23-s + (−0.923 + 0.382i)25-s + (0.324 − 0.216i)29-s + (1.63 − 0.324i)37-s + (0.923 − 1.38i)43-s + (−0.707 + 0.707i)49-s + (0.923 + 0.617i)53-s + 63-s + (0.617 + 0.923i)67-s + (−0.541 − 1.30i)71-s + (0.382 − 1.92i)77-s + (−1.30 + 1.30i)79-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)7-s + (−0.382 + 0.923i)9-s + (1.63 + 1.08i)11-s + (0.707 + 0.292i)23-s + (−0.923 + 0.382i)25-s + (0.324 − 0.216i)29-s + (1.63 − 0.324i)37-s + (0.923 − 1.38i)43-s + (−0.707 + 0.707i)49-s + (0.923 + 0.617i)53-s + 63-s + (0.617 + 0.923i)67-s + (−0.541 − 1.30i)71-s + (0.382 − 1.92i)77-s + (−1.30 + 1.30i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.956 - 0.290i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1553, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ 0.956 - 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.161558002\)
\(L(\frac12)\) \(\approx\) \(1.161558002\)
\(L(1)\) 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 \( 1 + (0.382 + 0.923i)T \)
good3 \( 1 + (0.382 - 0.923i)T^{2} \)
5 \( 1 + (0.923 - 0.382i)T^{2} \)
11 \( 1 + (-1.63 - 1.08i)T + (0.382 + 0.923i)T^{2} \)
13 \( 1 + (0.923 + 0.382i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-0.923 - 0.382i)T^{2} \)
23 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-1.63 + 0.324i)T + (0.923 - 0.382i)T^{2} \)
41 \( 1 + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + (-0.923 + 1.38i)T + (-0.382 - 0.923i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-0.923 - 0.617i)T + (0.382 + 0.923i)T^{2} \)
59 \( 1 + (0.923 - 0.382i)T^{2} \)
61 \( 1 + (0.382 - 0.923i)T^{2} \)
67 \( 1 + (-0.617 - 0.923i)T + (-0.382 + 0.923i)T^{2} \)
71 \( 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
83 \( 1 + (-0.923 - 0.382i)T^{2} \)
89 \( 1 + (-0.707 + 0.707i)T^{2} \)
97 \( 1 + T^{2} \)
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

−9.532195988687775325419346078309, −8.859887245696887159935920121430, −7.72018436645580173359276164721, −7.21932238641797150788291584663, −6.43872085052859933609468504443, −5.47082460797607461617844902079, −4.35241890283861609589443543987, −3.87812391465162544664653689493, −2.52620547517382661211322628620, −1.33490287060637897343076651866, 1.07440512623192178225494739388, 2.65546481801332640547621893441, 3.47249608988627339979921558485, 4.34318542016166654093871642496, 5.70231136970826123312882161984, 6.20113256823655667553079997485, 6.77203546130765159114518771094, 8.073529029115961106450433140040, 8.844232442038079942687742337025, 9.249981920833421041381667447467

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