↑

Properties

Label 2-792-8.5-c1-0-9
Degree $2$
Conductor $792$
Sign $-0.0424 - 0.999i$
Analytic cond. $6.32415$
Root an. cond. $2.51478$
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  + (−0.689 + 1.23i)2-s + (−1.04 − 1.70i)4-s − 0.752i·5-s − 3.95·7-s + (2.82 − 0.120i)8-s + (0.928 + 0.518i)10-s − i·11-s + 0.420i·13-s + (2.72 − 4.88i)14-s + (−1.80 + 3.57i)16-s + 3.06·17-s + 7.03i·19-s + (−1.28 + 0.788i)20-s + (1.23 + 0.689i)22-s + 3.63·23-s + ⋯
L(s)  = 1  + (−0.487 + 0.873i)2-s + (−0.524 − 0.851i)4-s − 0.336i·5-s − 1.49·7-s + (0.999 − 0.0424i)8-s + (0.293 + 0.164i)10-s − 0.301i·11-s + 0.116i·13-s + (0.729 − 1.30i)14-s + (−0.450 + 0.892i)16-s + 0.743·17-s + 1.61i·19-s + (−0.286 + 0.176i)20-s + (0.263 + 0.147i)22-s + 0.757·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0424 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0424 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $-0.0424 - 0.999i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ -0.0424 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.580764 + 0.605971i\)
\(L(\frac12)\) \(\approx\) \(0.580764 + 0.605971i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.689 - 1.23i)T \)
3 \( 1 \)
11 \( 1 + iT \)
good5 \( 1 + 0.752iT - 5T^{2} \)
7 \( 1 + 3.95T + 7T^{2} \)
13 \( 1 - 0.420iT - 13T^{2} \)
17 \( 1 - 3.06T + 17T^{2} \)
19 \( 1 - 7.03iT - 19T^{2} \)
23 \( 1 - 3.63T + 23T^{2} \)
29 \( 1 - 2.98iT - 29T^{2} \)
31 \( 1 + 2.93T + 31T^{2} \)
37 \( 1 - 1.20iT - 37T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 - 10.2iT - 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 - 8.38iT - 53T^{2} \)
59 \( 1 - 1.21iT - 59T^{2} \)
61 \( 1 + 3.86iT - 61T^{2} \)
67 \( 1 + 7.79iT - 67T^{2} \)
71 \( 1 + 6.91T + 71T^{2} \)
73 \( 1 + 2.76T + 73T^{2} \)
79 \( 1 + 6.67T + 79T^{2} \)
83 \( 1 - 14.8iT - 83T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 - 3.22T + 97T^{2} \)
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

−10.21603723053156896863483478903, −9.471946110381890745880220881789, −8.894072058432358292016092152653, −7.87628258067826584863344733139, −7.06176336955739959008807867674, −6.12257290969714667527752420416, −5.56275602759658295297855255909, −4.24226270999434029295449400497, −3.08737340248687110143486744691, −1.10374527515729095638114501487, 0.61273209477364166702183137540, 2.52679340719594952121103479976, 3.20963036035025273704013298126, 4.28704461892579583807870159684, 5.59027016065204568699287303289, 6.91734296526299750754701664979, 7.35786007270918450370098005407, 8.790776307422452617143282762515, 9.283531228668512107689547386374, 10.11325082424117836288199716244

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