↑

Properties

Label 2-471-157.156-c3-0-9
Degree $2$
Conductor $471$
Sign $-0.252 - 0.967i$
Analytic cond. $27.7898$
Root an. cond. $5.27161$
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  − 3.30i·2-s − 3·3-s − 2.91·4-s + 13.7i·5-s + 9.91i·6-s + 33.0i·7-s − 16.7i·8-s + 9·9-s + 45.3·10-s + 17.8·11-s + 8.75·12-s + 14.1·13-s + 109.·14-s − 41.2i·15-s − 78.8·16-s − 19.3·17-s + ⋯
L(s)  = 1  − 1.16i·2-s − 0.577·3-s − 0.364·4-s + 1.22i·5-s + 0.674i·6-s + 1.78i·7-s − 0.742i·8-s + 0.333·9-s + 1.43·10-s + 0.488·11-s + 0.210·12-s + 0.301·13-s + 2.08·14-s − 0.709i·15-s − 1.23·16-s − 0.276·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-0.252 - 0.967i$
Analytic conductor: \(27.7898\)
Root analytic conductor: \(5.27161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :3/2),\ -0.252 - 0.967i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7491004784\)
\(L(\frac12)\) \(\approx\) \(0.7491004784\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
157 \( 1 + (497. + 1.90e3i)T \)
good2 \( 1 + 3.30iT - 8T^{2} \)
5 \( 1 - 13.7iT - 125T^{2} \)
7 \( 1 - 33.0iT - 343T^{2} \)
11 \( 1 - 17.8T + 1.33e3T^{2} \)
13 \( 1 - 14.1T + 2.19e3T^{2} \)
17 \( 1 + 19.3T + 4.91e3T^{2} \)
19 \( 1 + 93.0T + 6.85e3T^{2} \)
23 \( 1 + 92.0iT - 1.21e4T^{2} \)
29 \( 1 - 264. iT - 2.43e4T^{2} \)
31 \( 1 + 67.0T + 2.97e4T^{2} \)
37 \( 1 + 196.T + 5.06e4T^{2} \)
41 \( 1 - 97.6iT - 6.89e4T^{2} \)
43 \( 1 + 36.0iT - 7.95e4T^{2} \)
47 \( 1 + 273.T + 1.03e5T^{2} \)
53 \( 1 + 260. iT - 1.48e5T^{2} \)
59 \( 1 + 669. iT - 2.05e5T^{2} \)
61 \( 1 + 266. iT - 2.26e5T^{2} \)
67 \( 1 + 97.0T + 3.00e5T^{2} \)
71 \( 1 - 338.T + 3.57e5T^{2} \)
73 \( 1 - 518. iT - 3.89e5T^{2} \)
79 \( 1 + 32.8iT - 4.93e5T^{2} \)
83 \( 1 - 1.32e3iT - 5.71e5T^{2} \)
89 \( 1 + 66.9T + 7.04e5T^{2} \)
97 \( 1 + 605. iT - 9.12e5T^{2} \)
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

−11.03991998259352187186443942547, −10.34039737525907787720745563678, −9.326469029023895290720116317996, −8.477921325358117353060954487955, −6.72183156995529965406497599041, −6.43794693696553920118275020190, −5.11451888433416092469888373899, −3.63409213192321797787933812133, −2.65809877277141728806730111047, −1.78865786338696647214106543996, 0.24875951499685661902883715101, 1.51685810166916877447343984355, 4.04234703444896963172258663970, 4.66206030526550261834750919774, 5.78331876223966897440424568189, 6.65854463630286668534315178571, 7.46013668760740900149828806700, 8.281519004182259063426807524681, 9.232333931688415524375627946469, 10.37818951164065606674760390448

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