↑

Properties

Label 2-525-21.17-c1-0-1
Degree $2$
Conductor $525$
Sign $-0.591 - 0.806i$
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  + (−0.334 + 0.192i)2-s + (0.139 − 1.72i)3-s + (−0.925 + 1.60i)4-s + (0.286 + 0.603i)6-s + (−2.36 − 1.17i)7-s − 1.48i·8-s + (−2.96 − 0.480i)9-s + (2.20 + 1.27i)11-s + (2.63 + 1.82i)12-s + 3.06i·13-s + (1.01 − 0.0640i)14-s + (−1.56 − 2.71i)16-s + (−3.23 + 5.59i)17-s + (1.08 − 0.410i)18-s + (−1.03 + 0.597i)19-s + ⋯
L(s)  = 1  + (−0.236 + 0.136i)2-s + (0.0803 − 0.996i)3-s + (−0.462 + 0.801i)4-s + (0.116 + 0.246i)6-s + (−0.895 − 0.444i)7-s − 0.525i·8-s + (−0.987 − 0.160i)9-s + (0.663 + 0.383i)11-s + (0.761 + 0.525i)12-s + 0.850i·13-s + (0.272 − 0.0171i)14-s + (−0.391 − 0.677i)16-s + (−0.783 + 1.35i)17-s + (0.254 − 0.0968i)18-s + (−0.237 + 0.137i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.591 - 0.806i$
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: \((2,\ 525,\ (\ :1/2),\ -0.591 - 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.166831 + 0.329212i\)
\(L(\frac12)\) \(\approx\) \(0.166831 + 0.329212i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.139 + 1.72i)T \)
5 \( 1 \)
7 \( 1 + (2.36 + 1.17i)T \)
good2 \( 1 + (0.334 - 0.192i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-2.20 - 1.27i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.06iT - 13T^{2} \)
17 \( 1 + (3.23 - 5.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.03 - 0.597i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.64 - 1.52i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.77iT - 29T^{2} \)
31 \( 1 + (5.95 + 3.43i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.77 + 3.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.31T + 41T^{2} \)
43 \( 1 + 5.46T + 43T^{2} \)
47 \( 1 + (1.61 + 2.78i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-11.4 - 6.62i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.98 + 3.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.08 - 4.67i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.75 - 3.04i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.921iT - 71T^{2} \)
73 \( 1 + (0.256 + 0.148i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.14 + 7.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.11T + 83T^{2} \)
89 \( 1 + (9.41 + 16.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.3iT - 97T^{2} \)
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

−11.35201293488656747838047373154, −10.15789726759036361307970259532, −9.027463713057707018894130578254, −8.637579825750240660450550121193, −7.37145349934355974883807226632, −6.89972936635025021202644457828, −5.98949096648217939574206626865, −4.23308918849880942085159989703, −3.41783692477824567712954318074, −1.81293723027543915761514149399, 0.22480929405697082734978097949, 2.53393795983369730489632662016, 3.75590373970217208145125122854, 4.91435567590478837886438148735, 5.74308687827467107866988524305, 6.64000993144877388479602248208, 8.313945439500158277449099453344, 9.061434189762599569556093359132, 9.659088836040314992484239336019, 10.32688898181906016459793305635

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