↑

Properties

Label 2-522-29.24-c1-0-9
Degree $2$
Conductor $522$
Sign $0.995 + 0.0992i$
Analytic cond. $4.16819$
Root an. cond. $2.04161$
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  + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)4-s + (−0.969 − 1.21i)5-s + (−0.777 − 3.40i)7-s + (−0.900 + 0.433i)8-s + (0.346 − 1.51i)10-s + (3.37 + 1.62i)11-s + (3.46 + 1.67i)13-s + (2.17 − 2.73i)14-s + (−0.900 − 0.433i)16-s + 4.93·17-s + (1.37 − 6.00i)19-s + (1.40 − 0.674i)20-s + (0.832 + 3.64i)22-s + (3.59 − 4.50i)23-s + ⋯
L(s)  = 1  + (0.440 + 0.552i)2-s + (−0.111 + 0.487i)4-s + (−0.433 − 0.543i)5-s + (−0.293 − 1.28i)7-s + (−0.318 + 0.153i)8-s + (0.109 − 0.479i)10-s + (1.01 + 0.489i)11-s + (0.962 + 0.463i)13-s + (0.582 − 0.730i)14-s + (−0.225 − 0.108i)16-s + 1.19·17-s + (0.314 − 1.37i)19-s + (0.313 − 0.150i)20-s + (0.177 + 0.777i)22-s + (0.749 − 0.939i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(522\)    =    \(2 \cdot 3^{2} \cdot 29\)
Sign: $0.995 + 0.0992i$
Analytic conductor: \(4.16819\)
Root analytic conductor: \(2.04161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{522} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 522,\ (\ :1/2),\ 0.995 + 0.0992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72668 - 0.0859099i\)
\(L(\frac12)\) \(\approx\) \(1.72668 - 0.0859099i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.623 - 0.781i)T \)
3 \( 1 \)
29 \( 1 + (5.38 - 0.202i)T \)
good5 \( 1 + (0.969 + 1.21i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (0.777 + 3.40i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (-3.37 - 1.62i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-3.46 - 1.67i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 - 4.93T + 17T^{2} \)
19 \( 1 + (-1.37 + 6.00i)T + (-17.1 - 8.24i)T^{2} \)
23 \( 1 + (-3.59 + 4.50i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (-1.56 - 1.96i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 + (6.11 - 2.94i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + 0.0271T + 41T^{2} \)
43 \( 1 + (7.32 - 9.17i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (0.5 + 0.240i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (2.03 + 2.54i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 - 3.06T + 59T^{2} \)
61 \( 1 + (0.0293 + 0.128i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (-9.91 + 4.77i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (3.26 + 1.57i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (4.13 - 5.18i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + (1.35 - 0.653i)T + (49.2 - 61.7i)T^{2} \)
83 \( 1 + (2.21 - 9.71i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-3.36 - 4.21i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (0.960 - 4.20i)T + (-87.3 - 42.0i)T^{2} \)
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

−10.99448201011719271342037866030, −9.869969753243760218474357430956, −8.946988412837015823541562028665, −8.067533351907846016877606291743, −6.98387913247274887258517369460, −6.55154736015908672410767778820, −5.02778699158858421594893339110, −4.20750436449655041698427013196, −3.36949649506379339597176538873, −1.05964273498067671744030937348, 1.57598416938191607379301000214, 3.30879586800955837209155838166, 3.60320425775720816576666937535, 5.52309200411390711298099106990, 5.88598629847022824625512500690, 7.17673594871081563917949800254, 8.376553138955197058036296293873, 9.195353403941229451318696205708, 10.09068824963200107096828046727, 11.13120431795038760073768352090

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