↑

Properties

Label 2-525-21.17-c1-0-42
Degree $2$
Conductor $525$
Sign $0.299 + 0.953i$
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  + (2.17 − 1.25i)2-s + (1.73 − 0.0401i)3-s + (2.14 − 3.71i)4-s + (3.71 − 2.25i)6-s + (−2.06 + 1.65i)7-s − 5.74i·8-s + (2.99 − 0.138i)9-s + (−1.48 − 0.859i)11-s + (3.56 − 6.51i)12-s + 0.360i·13-s + (−2.39 + 6.18i)14-s + (−2.91 − 5.04i)16-s + (−1.27 + 2.20i)17-s + (6.33 − 4.05i)18-s + (−4.93 + 2.84i)19-s + ⋯
L(s)  = 1  + (1.53 − 0.886i)2-s + (0.999 − 0.0231i)3-s + (1.07 − 1.85i)4-s + (1.51 − 0.922i)6-s + (−0.778 + 0.627i)7-s − 2.03i·8-s + (0.998 − 0.0463i)9-s + (−0.448 − 0.259i)11-s + (1.02 − 1.88i)12-s + 0.100i·13-s + (−0.639 + 1.65i)14-s + (−0.727 − 1.26i)16-s + (−0.308 + 0.534i)17-s + (1.49 − 0.956i)18-s + (−1.13 + 0.653i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.299 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.299 + 0.953i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.299 + 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.27675 - 2.40492i\)
\(L(\frac12)\) \(\approx\) \(3.27675 - 2.40492i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 + 0.0401i)T \)
5 \( 1 \)
7 \( 1 + (2.06 - 1.65i)T \)
good2 \( 1 + (-2.17 + 1.25i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (1.48 + 0.859i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.360iT - 13T^{2} \)
17 \( 1 + (1.27 - 2.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.93 - 2.84i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.17 + 1.25i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.76iT - 29T^{2} \)
31 \( 1 + (2.41 + 1.39i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.65 - 2.86i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.63T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 + (-2.91 - 5.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.25 - 0.727i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.42 - 5.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.38 + 0.801i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.24 + 2.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.1iT - 71T^{2} \)
73 \( 1 + (-10.0 - 5.82i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.93 + 12.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.50T + 83T^{2} \)
89 \( 1 + (6.10 + 10.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.18iT - 97T^{2} \)
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

−10.73980165761843680123791512421, −10.08102621819846121391926308910, −9.074678255009932010821176088701, −8.079006231100045215289956685397, −6.63117014323239046183780987605, −5.89331200067215382736564761518, −4.63204796451176986532227374238, −3.70999783835612700419209549471, −2.82457394652934914484305455289, −1.94057654447273535633215103819, 2.51458170973122781930858160858, 3.46820300929947064145412225328, 4.32609371000911298151044420812, 5.26996317848594373515497977912, 6.69954678419089114741643049721, 7.04490837477303716312941999736, 8.024531091893439168422227140763, 9.047700381337351500507089299426, 10.12901866831140731581980728578, 11.19180758275850718937399546751

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