↑

Properties

Label 2-525-5.4-c3-0-40
Degree $2$
Conductor $525$
Sign $-0.447 + 0.894i$
Analytic cond. $30.9760$
Root an. cond. $5.56560$
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  − 1.70i·2-s − 3i·3-s + 5.10·4-s − 5.10·6-s + 7i·7-s − 22.2i·8-s − 9·9-s + 37.4·11-s − 15.3i·12-s − 29.0i·13-s + 11.9·14-s + 2.89·16-s + 58.4i·17-s + 15.3i·18-s + 54.5·19-s + ⋯
L(s)  = 1  − 0.601i·2-s − 0.577i·3-s + 0.638·4-s − 0.347·6-s + 0.377i·7-s − 0.985i·8-s − 0.333·9-s + 1.02·11-s − 0.368i·12-s − 0.619i·13-s + 0.227·14-s + 0.0452·16-s + 0.833i·17-s + 0.200i·18-s + 0.659·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(30.9760\)
Root analytic conductor: \(5.56560\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :3/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.483173384\)
\(L(\frac12)\) \(\approx\) \(2.483173384\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3iT \)
5 \( 1 \)
7 \( 1 - 7iT \)
good2 \( 1 + 1.70iT - 8T^{2} \)
11 \( 1 - 37.4T + 1.33e3T^{2} \)
13 \( 1 + 29.0iT - 2.19e3T^{2} \)
17 \( 1 - 58.4iT - 4.91e3T^{2} \)
19 \( 1 - 54.5T + 6.85e3T^{2} \)
23 \( 1 + 161. iT - 1.21e4T^{2} \)
29 \( 1 + 137.T + 2.43e4T^{2} \)
31 \( 1 - 154.T + 2.97e4T^{2} \)
37 \( 1 + 350. iT - 5.06e4T^{2} \)
41 \( 1 - 353.T + 6.89e4T^{2} \)
43 \( 1 - 518. iT - 7.95e4T^{2} \)
47 \( 1 + 542. iT - 1.03e5T^{2} \)
53 \( 1 + 305. iT - 1.48e5T^{2} \)
59 \( 1 + 14.6T + 2.05e5T^{2} \)
61 \( 1 + 171.T + 2.26e5T^{2} \)
67 \( 1 - 551. iT - 3.00e5T^{2} \)
71 \( 1 + 120.T + 3.57e5T^{2} \)
73 \( 1 + 284. iT - 3.89e5T^{2} \)
79 \( 1 + 941.T + 4.93e5T^{2} \)
83 \( 1 + 377. iT - 5.71e5T^{2} \)
89 \( 1 - 677.T + 7.04e5T^{2} \)
97 \( 1 + 1.22e3iT - 9.12e5T^{2} \)
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

−10.34077258387896157738125198695, −9.379700224307719900981578086573, −8.369270503816395697422366970009, −7.38774583064520151755923677994, −6.46760973555944416210231163172, −5.75731937302738976463607514912, −4.12312531176882143638939626380, −2.98568718307386687278641873315, −1.94162757884440659143311000644, −0.798245066365011784058325706290, 1.40047953370695651041519233341, 2.93953450709289900493870022123, 4.10584405886855613180340072901, 5.24277622308417801666530011906, 6.21225056256950016229078713362, 7.10623925716719638895432003291, 7.83092551379854683403825392977, 9.088882900762429916558496784427, 9.672263532239419960032676827589, 10.83680943085414779157017277532

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