Properties

Label 2-525-105.104-c1-0-24
Degree $2$
Conductor $525$
Sign $-0.353 + 0.935i$
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  − 0.792·2-s + (−1.26 − 1.18i)3-s − 1.37·4-s + (1 + 0.939i)6-s + (1.73 − 2i)7-s + 2.67·8-s + (0.186 + 2.99i)9-s + 2.52i·11-s + (1.73 + 1.62i)12-s + 4.10·13-s + (−1.37 + 1.58i)14-s + 0.627·16-s − 4.37i·17-s + (−0.147 − 2.37i)18-s − 3.46i·19-s + ⋯
L(s)  = 1  − 0.560·2-s + (−0.728 − 0.684i)3-s − 0.686·4-s + (0.408 + 0.383i)6-s + (0.654 − 0.755i)7-s + 0.944·8-s + (0.0620 + 0.998i)9-s + 0.761i·11-s + (0.499 + 0.469i)12-s + 1.13·13-s + (−0.366 + 0.423i)14-s + 0.156·16-s − 1.06i·17-s + (−0.0347 − 0.559i)18-s − 0.794i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.353 + 0.935i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.353 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.360603 - 0.521552i\)
\(L(\frac12)\) \(\approx\) \(0.360603 - 0.521552i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.26 + 1.18i)T \)
5 \( 1 \)
7 \( 1 + (-1.73 + 2i)T \)
good2 \( 1 + 0.792T + 2T^{2} \)
11 \( 1 - 2.52iT - 11T^{2} \)
13 \( 1 - 4.10T + 13T^{2} \)
17 \( 1 + 4.37iT - 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 8.51T + 23T^{2} \)
29 \( 1 - 0.939iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + 6.74iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4.74iT - 43T^{2} \)
47 \( 1 - 1.62iT - 47T^{2} \)
53 \( 1 - 1.87T + 53T^{2} \)
59 \( 1 + 8.74T + 59T^{2} \)
61 \( 1 + 6.92iT - 61T^{2} \)
67 \( 1 - 4.74iT - 67T^{2} \)
71 \( 1 + 0.294iT - 71T^{2} \)
73 \( 1 + 6.92T + 73T^{2} \)
79 \( 1 - 2.37T + 79T^{2} \)
83 \( 1 + 17.4iT - 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + 11.0T + 97T^{2} \)
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

−10.63988870759656665682750013300, −9.772662217233588073699929816884, −8.738623351911615429687544527529, −7.73665478657744840759737994714, −7.27394092525433108078422570712, −6.00420544996211505770274477820, −4.88141905055264478122605991309, −4.08953277855884655382921073590, −1.87915412785162903504348455049, −0.57372402697817544619201336355, 1.39317471071767236800489210447, 3.62335426485669827650163998968, 4.47397752291348499491910428551, 5.68438275121213461960748691131, 6.16920646311500701615969721933, 8.050058037600516035206703354299, 8.470907394253547945812791848650, 9.367294081354331639314546916263, 10.30890505654421235386925231836, 10.89293634651144708243326705064

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