↑

Properties

Label 2-525-21.20-c1-0-14
Degree $2$
Conductor $525$
Sign $-0.885 - 0.464i$
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  + 2.52i·2-s + (1.68 − 0.396i)3-s − 4.37·4-s + (1 + 4.25i)6-s + (2 + 1.73i)7-s − 5.98i·8-s + (2.68 − 1.33i)9-s − 0.792i·11-s + (−7.37 + 1.73i)12-s + 5.84i·13-s + (−4.37 + 5.04i)14-s + 6.37·16-s − 1.37·17-s + (3.37 + 6.78i)18-s + 3.46i·19-s + ⋯
L(s)  = 1  + 1.78i·2-s + (0.973 − 0.228i)3-s − 2.18·4-s + (0.408 + 1.73i)6-s + (0.755 + 0.654i)7-s − 2.11i·8-s + (0.895 − 0.445i)9-s − 0.238i·11-s + (−2.12 + 0.500i)12-s + 1.61i·13-s + (−1.16 + 1.34i)14-s + 1.59·16-s − 0.332·17-s + (0.794 + 1.59i)18-s + 0.794i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 - 0.464i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.885 - 0.464i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.885 - 0.464i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.441581 + 1.79293i\)
\(L(\frac12)\) \(\approx\) \(0.441581 + 1.79293i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.68 + 0.396i)T \)
5 \( 1 \)
7 \( 1 + (-2 - 1.73i)T \)
good2 \( 1 - 2.52iT - 2T^{2} \)
11 \( 1 + 0.792iT - 11T^{2} \)
13 \( 1 - 5.84iT - 13T^{2} \)
17 \( 1 + 1.37T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 - 1.87iT - 23T^{2} \)
29 \( 1 + 4.25iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + 4.74T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 6.74T + 43T^{2} \)
47 \( 1 + 7.37T + 47T^{2} \)
53 \( 1 + 8.51iT - 53T^{2} \)
59 \( 1 + 2.74T + 59T^{2} \)
61 \( 1 + 6.92iT - 61T^{2} \)
67 \( 1 - 6.74T + 67T^{2} \)
71 \( 1 + 13.5iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 - 3.37T + 79T^{2} \)
83 \( 1 - 5.48T + 83T^{2} \)
89 \( 1 + 3.25T + 89T^{2} \)
97 \( 1 + 1.08iT - 97T^{2} \)
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

−11.30740844567431670157672959784, −9.647905427679683335725715223337, −9.110491063056126051868608100519, −8.310798966662436835485204978103, −7.73154494346181329696825614827, −6.78089032033873958235922378778, −5.97221191883742469031545180874, −4.77007774029909082522363410137, −3.89878426004918868454145029312, −1.99154197073357210230909103551, 1.11413937516957643838729270996, 2.43568189146086026795211799547, 3.32478222341995324758066959219, 4.34321259098744994620593638086, 5.13371807346298224886706808717, 7.23967792837643326161007567616, 8.216494757473586717127498286070, 8.928713549598253981630344896422, 9.862605307066389210997403326218, 10.67079170619432034982626736392

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