↑

Properties

Label 2-525-35.33-c1-0-14
Degree $2$
Conductor $525$
Sign $0.957 - 0.289i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
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  + (0.401 + 1.49i)2-s + (−0.965 − 0.258i)3-s + (−0.346 + 0.200i)4-s − 1.54i·6-s + (1.01 − 2.44i)7-s + (1.75 + 1.75i)8-s + (0.866 + 0.499i)9-s + (−2.59 − 4.49i)11-s + (0.386 − 0.103i)12-s + (3.30 − 3.30i)13-s + (4.06 + 0.545i)14-s + (−2.32 + 4.01i)16-s + (−0.00519 + 0.0194i)17-s + (−0.401 + 1.49i)18-s + (1.24 − 2.14i)19-s + ⋯
L(s)  = 1  + (0.283 + 1.05i)2-s + (−0.557 − 0.149i)3-s + (−0.173 + 0.100i)4-s − 0.632i·6-s + (0.385 − 0.922i)7-s + (0.619 + 0.619i)8-s + (0.288 + 0.166i)9-s + (−0.782 − 1.35i)11-s + (0.111 − 0.0299i)12-s + (0.917 − 0.917i)13-s + (1.08 + 0.145i)14-s + (−0.580 + 1.00i)16-s + (−0.00126 + 0.00470i)17-s + (−0.0945 + 0.352i)18-s + (0.284 − 0.492i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.957 - 0.289i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.957 - 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58218 + 0.233722i\)
\(L(\frac12)\) \(\approx\) \(1.58218 + 0.233722i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 + (-1.01 + 2.44i)T \)
good2 \( 1 + (-0.401 - 1.49i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (2.59 + 4.49i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.30 + 3.30i)T - 13iT^{2} \)
17 \( 1 + (0.00519 - 0.0194i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.24 + 2.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.24 + 0.601i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 10.2iT - 29T^{2} \)
31 \( 1 + (-5.69 + 3.28i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.714 - 2.66i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 3.68iT - 41T^{2} \)
43 \( 1 + (-2.79 - 2.79i)T + 43iT^{2} \)
47 \( 1 + (1.13 - 0.303i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.23 - 4.60i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.222 + 0.385i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.18 - 0.684i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.70 + 1.52i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 2.14T + 71T^{2} \)
73 \( 1 + (7.15 + 1.91i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.47 + 2.00i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.77 + 3.77i)T - 83iT^{2} \)
89 \( 1 + (-1.91 + 3.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.5 + 10.5i)T + 97iT^{2} \)
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

−10.81085946636906276292222748638, −10.44982134320375035896618320521, −8.704287478828508162463489243662, −7.956259434854708785266211311225, −7.20069187420398063922687483844, −6.23238079269782083948003234205, −5.50295249082494738829404176066, −4.65252042137686381677415353687, −3.16677810608069359390868924753, −1.04202239157763727627591146658, 1.62792627663100626511061510034, 2.63436419284832668294490198585, 4.09869934846386532144546044681, 4.88238319235218714332861950376, 6.05754734795749117049415598170, 7.13776437356453042105067829274, 8.192319736047590322799263838380, 9.462483633015679856458971722390, 10.10747948114120408912516973484, 11.04260625570751452577942817189

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