↑

Properties

Label 2-525-105.89-c1-0-10
Degree $2$
Conductor $525$
Sign $-0.602 - 0.797i$
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  + (0.866 + 1.5i)3-s + (1 + 1.73i)4-s + (−2.59 + 0.5i)7-s + (−1.5 + 2.59i)9-s + (−1.73 + 3i)12-s − 1.73·13-s + (−1.99 + 3.46i)16-s + (4.5 + 2.59i)19-s + (−3 − 3.46i)21-s − 5.19·27-s + (−3.46 − 4i)28-s + (7.5 − 4.33i)31-s − 6·36-s + (−0.866 − 0.5i)37-s + (−1.49 − 2.59i)39-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.981 + 0.188i)7-s + (−0.5 + 0.866i)9-s + (−0.499 + 0.866i)12-s − 0.480·13-s + (−0.499 + 0.866i)16-s + (1.03 + 0.596i)19-s + (−0.654 − 0.755i)21-s − 1.00·27-s + (−0.654 − 0.755i)28-s + (1.34 − 0.777i)31-s − 36-s + (−0.142 − 0.0821i)37-s + (−0.240 − 0.416i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.602 - 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.602 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.602 - 0.797i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.602 - 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.682501 + 1.37086i\)
\(L(\frac12)\) \(\approx\) \(0.682501 + 1.37086i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 \)
7 \( 1 + (2.59 - 0.5i)T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.73T + 13T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.5 - 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-7.5 + 4.33i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 5iT - 43T^{2} \)
47 \( 1 + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 - 3.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.52 + 5.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-7.79 - 13.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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

−11.19428633243708219733248523818, −10.04597020844232691814020991386, −9.555060124826603028300992549301, −8.490157019605125841283490519674, −7.73173825126226553397471822905, −6.71237211341221320172796152755, −5.56344451071296953292575847867, −4.22955534082884684489211131609, −3.29437117069701843285306384898, −2.49114623378928805010886346824, 0.836185044095732714135940194142, 2.35076606255074467952246743935, 3.33780499416455330522118645996, 5.07277636183425277031530481028, 6.20631761641437912683384100716, 6.85760136313282578749802655242, 7.59936182012340798561216368188, 8.886953193073186129762757045930, 9.672412624871285420971848158922, 10.39837357569886412223516445746

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