↑

Properties

Label 2-525-5.4-c1-0-8
Degree $2$
Conductor $525$
Sign $-0.447 + 0.894i$
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  − 2.23i·2-s + i·3-s − 3.00·4-s + 2.23·6-s + i·7-s + 2.23i·8-s − 9-s + 6.47·11-s − 3.00i·12-s − 4.47i·13-s + 2.23·14-s − 0.999·16-s − 2i·17-s + 2.23i·18-s + 2.47·19-s + ⋯
L(s)  = 1  − 1.58i·2-s + 0.577i·3-s − 1.50·4-s + 0.912·6-s + 0.377i·7-s + 0.790i·8-s − 0.333·9-s + 1.95·11-s − 0.866i·12-s − 1.24i·13-s + 0.597·14-s − 0.249·16-s − 0.485i·17-s + 0.527i·18-s + 0.567·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.753019 - 1.21841i\)
\(L(\frac12)\) \(\approx\) \(0.753019 - 1.21841i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 \)
7 \( 1 - iT \)
good2 \( 1 + 2.23iT - 2T^{2} \)
11 \( 1 - 6.47T + 11T^{2} \)
13 \( 1 + 4.47iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 2.47T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 1.52T + 31T^{2} \)
37 \( 1 + 6.94iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 8.94iT - 43T^{2} \)
47 \( 1 - 12.9iT - 47T^{2} \)
53 \( 1 - 3.52iT - 53T^{2} \)
59 \( 1 - 8.94T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 5.52T + 71T^{2} \)
73 \( 1 - 12.4iT - 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 - 16.9iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 8.47iT - 97T^{2} \)
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

−10.69159660920338494318787643276, −9.783530622175747585537367935126, −9.226574554431878718800973910902, −8.407257462357566950335222864294, −6.86562297333964146479448047341, −5.60923465722218931541463447817, −4.40948195333990313019834533205, −3.56652838186879732080838600023, −2.57591457004942958066563330914, −0.999023699451648926557985551450, 1.52350592846779138233653270045, 3.74399786939497344700652267645, 4.75268583486554641602332119434, 6.05455873304888378213257825694, 6.68814616729191600085892820071, 7.21935505245924317746793034233, 8.313458010137320675761846350984, 9.024912832074139231390662548731, 9.835646425596061911027474727899, 11.53389716584980620743750507453

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