↑

Properties

Label 2-525-35.19-c2-0-22
Degree $2$
Conductor $525$
Sign $0.625 - 0.780i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.18 + 1.26i)2-s + (−0.866 − 1.5i)3-s + (1.17 + 2.03i)4-s − 4.36i·6-s + (−3.28 + 6.18i)7-s − 4.15i·8-s + (−1.5 + 2.59i)9-s + (4.36 + 7.55i)11-s + (2.03 − 3.52i)12-s + 21.5·13-s + (−14.9 + 9.34i)14-s + (9.93 − 17.2i)16-s + (10.8 + 18.7i)17-s + (−6.54 + 3.78i)18-s + (2.71 + 1.56i)19-s + ⋯
L(s)  = 1  + (1.09 + 0.630i)2-s + (−0.288 − 0.5i)3-s + (0.294 + 0.509i)4-s − 0.727i·6-s + (−0.469 + 0.882i)7-s − 0.519i·8-s + (−0.166 + 0.288i)9-s + (0.396 + 0.686i)11-s + (0.169 − 0.294i)12-s + 1.65·13-s + (−1.06 + 0.667i)14-s + (0.621 − 1.07i)16-s + (0.638 + 1.10i)17-s + (−0.363 + 0.210i)18-s + (0.142 + 0.0825i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.625 - 0.780i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ 0.625 - 0.780i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.884913666\)
\(L(\frac12)\) \(\approx\) \(2.884913666\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 1.5i)T \)
5 \( 1 \)
7 \( 1 + (3.28 - 6.18i)T \)
good2 \( 1 + (-2.18 - 1.26i)T + (2 + 3.46i)T^{2} \)
11 \( 1 + (-4.36 - 7.55i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 21.5T + 169T^{2} \)
17 \( 1 + (-10.8 - 18.7i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-2.71 - 1.56i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (3.55 + 2.05i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 50.8T + 841T^{2} \)
31 \( 1 + (33.9 - 19.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-45.8 - 26.4i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 36.8iT - 1.68e3T^{2} \)
43 \( 1 - 17.6iT - 1.84e3T^{2} \)
47 \( 1 + (2.01 - 3.49i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-3.85 + 2.22i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (81.5 - 47.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (63.3 + 36.5i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-87.0 + 50.2i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 56.6T + 5.04e3T^{2} \)
73 \( 1 + (-37.4 - 64.8i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-14.4 + 25.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 21.1T + 6.88e3T^{2} \)
89 \( 1 + (63.1 + 36.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 73.7T + 9.40e3T^{2} \)
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

−10.94073849372935120335780509282, −9.913176239382402498177631565423, −8.842279629301357517735056340455, −7.88712782512813589778744241158, −6.60450287326231924228482331540, −6.19830749746160993284446921268, −5.40023574005825213305687833776, −4.20258867175461126018840298385, −3.15386926440872421994054598964, −1.38916138298356743725912209225, 0.996567947887048197212949720418, 3.01521133878264097989249065294, 3.72122355462976601784647401383, 4.53797616265143089114881584808, 5.70450199433460058319778022870, 6.44059397684102577530236224760, 7.83344709749683458492221287760, 8.902419220377681694698586700132, 9.904611496923753504734040668192, 10.96815696073507027366344225604

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