Properties

Label 2-1792-1792.1021-c0-0-0
Degree $2$
Conductor $1792$
Sign $-0.757 - 0.653i$
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  + (0.471 + 0.881i)2-s + (−0.555 + 0.831i)4-s + (0.0980 + 0.995i)7-s + (−0.995 − 0.0980i)8-s + (0.634 + 0.773i)9-s + (−0.0238 − 0.485i)11-s + (−0.831 + 0.555i)14-s + (−0.382 − 0.923i)16-s + (−0.382 + 0.923i)18-s + (0.416 − 0.249i)22-s + (−0.728 + 1.36i)23-s + (−0.956 − 0.290i)25-s + (−0.881 − 0.471i)28-s + (1.32 + 1.46i)29-s + (0.634 − 0.773i)32-s + ⋯
L(s)  = 1  + (0.471 + 0.881i)2-s + (−0.555 + 0.831i)4-s + (0.0980 + 0.995i)7-s + (−0.995 − 0.0980i)8-s + (0.634 + 0.773i)9-s + (−0.0238 − 0.485i)11-s + (−0.831 + 0.555i)14-s + (−0.382 − 0.923i)16-s + (−0.382 + 0.923i)18-s + (0.416 − 0.249i)22-s + (−0.728 + 1.36i)23-s + (−0.956 − 0.290i)25-s + (−0.881 − 0.471i)28-s + (1.32 + 1.46i)29-s + (0.634 − 0.773i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.757 - 0.653i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ -0.757 - 0.653i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.303461373\)
\(L(\frac12)\) \(\approx\) \(1.303461373\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.471 - 0.881i)T \)
7 \( 1 + (-0.0980 - 0.995i)T \)
good3 \( 1 + (-0.634 - 0.773i)T^{2} \)
5 \( 1 + (0.956 + 0.290i)T^{2} \)
11 \( 1 + (0.0238 + 0.485i)T + (-0.995 + 0.0980i)T^{2} \)
13 \( 1 + (0.290 + 0.956i)T^{2} \)
17 \( 1 + (0.382 - 0.923i)T^{2} \)
19 \( 1 + (0.881 + 0.471i)T^{2} \)
23 \( 1 + (0.728 - 1.36i)T + (-0.555 - 0.831i)T^{2} \)
29 \( 1 + (-1.32 - 1.46i)T + (-0.0980 + 0.995i)T^{2} \)
31 \( 1 + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.0504 + 0.0841i)T + (-0.471 - 0.881i)T^{2} \)
41 \( 1 + (-0.831 + 0.555i)T^{2} \)
43 \( 1 + (0.633 + 1.33i)T + (-0.634 + 0.773i)T^{2} \)
47 \( 1 + (-0.923 - 0.382i)T^{2} \)
53 \( 1 + (-1.26 + 1.39i)T + (-0.0980 - 0.995i)T^{2} \)
59 \( 1 + (0.290 - 0.956i)T^{2} \)
61 \( 1 + (0.773 - 0.634i)T^{2} \)
67 \( 1 + (0.805 - 0.288i)T + (0.773 - 0.634i)T^{2} \)
71 \( 1 + (-0.301 - 0.247i)T + (0.195 + 0.980i)T^{2} \)
73 \( 1 + (0.980 + 0.195i)T^{2} \)
79 \( 1 + (-0.113 - 0.569i)T + (-0.923 + 0.382i)T^{2} \)
83 \( 1 + (0.471 - 0.881i)T^{2} \)
89 \( 1 + (0.555 - 0.831i)T^{2} \)
97 \( 1 + (-0.707 + 0.707i)T^{2} \)
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

−9.611456662091780465865658513352, −8.681718415170750627896242878179, −8.193087705912769708226919518816, −7.36510944462419062486255163552, −6.57164078122298421295102952652, −5.58856443221051770736516915086, −5.18775435097005200629370575759, −4.12350121849901264566584191723, −3.16268793168900885070547333671, −1.96631542608826617629716128767, 0.879490321285036588000163682783, 2.09790960382746045869135236633, 3.28440573487468933344923331038, 4.35670802343367072263063092762, 4.48347882530753672422103689553, 5.98562663589871429723258555280, 6.58414805329347771051174930455, 7.58748717650147223994580455971, 8.493629524779631927267466353758, 9.557570056640291054066143615319

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