↑

Properties

Label 2-567-21.20-c3-0-29
Degree $2$
Conductor $567$
Sign $0.274 - 0.961i$
Analytic cond. $33.4540$
Root an. cond. $5.78395$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.725i·2-s + 7.47·4-s − 11.0·5-s + (−17.8 − 5.08i)7-s + 11.2i·8-s − 8.03i·10-s + 0.253i·11-s − 14.3i·13-s + (3.69 − 12.9i)14-s + 51.6·16-s + 92.5·17-s − 130. i·19-s − 82.7·20-s − 0.184·22-s + 118. i·23-s + ⋯
L(s)  = 1  + 0.256i·2-s + 0.934·4-s − 0.989·5-s + (−0.961 − 0.274i)7-s + 0.496i·8-s − 0.253i·10-s + 0.00695i·11-s − 0.305i·13-s + (0.0705 − 0.246i)14-s + 0.806·16-s + 1.32·17-s − 1.57i·19-s − 0.924·20-s − 0.00178·22-s + 1.07i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.274 - 0.961i$
Analytic conductor: \(33.4540\)
Root analytic conductor: \(5.78395\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (566, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :3/2),\ 0.274 - 0.961i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.594812206\)
\(L(\frac12)\) \(\approx\) \(1.594812206\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (17.8 + 5.08i)T \)
good2 \( 1 - 0.725iT - 8T^{2} \)
5 \( 1 + 11.0T + 125T^{2} \)
11 \( 1 - 0.253iT - 1.33e3T^{2} \)
13 \( 1 + 14.3iT - 2.19e3T^{2} \)
17 \( 1 - 92.5T + 4.91e3T^{2} \)
19 \( 1 + 130. iT - 6.85e3T^{2} \)
23 \( 1 - 118. iT - 1.21e4T^{2} \)
29 \( 1 - 286. iT - 2.43e4T^{2} \)
31 \( 1 - 247. iT - 2.97e4T^{2} \)
37 \( 1 - 188.T + 5.06e4T^{2} \)
41 \( 1 - 106.T + 6.89e4T^{2} \)
43 \( 1 - 43.1T + 7.95e4T^{2} \)
47 \( 1 + 137.T + 1.03e5T^{2} \)
53 \( 1 - 419. iT - 1.48e5T^{2} \)
59 \( 1 + 435.T + 2.05e5T^{2} \)
61 \( 1 - 188. iT - 2.26e5T^{2} \)
67 \( 1 - 370.T + 3.00e5T^{2} \)
71 \( 1 - 26.1iT - 3.57e5T^{2} \)
73 \( 1 - 728. iT - 3.89e5T^{2} \)
79 \( 1 - 97.3T + 4.93e5T^{2} \)
83 \( 1 - 802.T + 5.71e5T^{2} \)
89 \( 1 + 236.T + 7.04e5T^{2} \)
97 \( 1 + 1.46e3iT - 9.12e5T^{2} \)
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

−10.67779382767919518646794928590, −9.700454681439519417083617248330, −8.637111001921455068373282123860, −7.46299252658248273604030883227, −7.19734167781931285633515224369, −6.10379968600953615252236844139, −5.04074205549157321119046521531, −3.52854410601951094443586817439, −2.93052197549353156785362070528, −1.07116770555049209018978810101, 0.54460341394867100907904110658, 2.20285763096561676109050402827, 3.36022468266813695993551637970, 4.09596676691420291461267174093, 5.85945189161034525338184662158, 6.40975857320262981815026929323, 7.69715083456899683578210409069, 8.021934677861556362752201773540, 9.602861649461896823777878528891, 10.11013774058752309848991535245

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