↑

Properties

Label 2-525-35.13-c1-0-23
Degree $2$
Conductor $525$
Sign $-0.777 + 0.629i$
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  + (1.68 − 1.68i)2-s + (0.707 − 0.707i)3-s − 3.66i·4-s − 2.38i·6-s + (−0.901 − 2.48i)7-s + (−2.81 − 2.81i)8-s − 1.00i·9-s + 1.54·11-s + (−2.59 − 2.59i)12-s + (−4.16 + 4.16i)13-s + (−5.70 − 2.66i)14-s − 2.12·16-s + (2.05 + 2.05i)17-s + (−1.68 − 1.68i)18-s + 5.70·19-s + ⋯
L(s)  = 1  + (1.19 − 1.19i)2-s + (0.408 − 0.408i)3-s − 1.83i·4-s − 0.972i·6-s + (−0.340 − 0.940i)7-s + (−0.993 − 0.993i)8-s − 0.333i·9-s + 0.465·11-s + (−0.748 − 0.748i)12-s + (−1.15 + 1.15i)13-s + (−1.52 − 0.713i)14-s − 0.530·16-s + (0.499 + 0.499i)17-s + (−0.396 − 0.396i)18-s + 1.30·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.777 + 0.629i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.777 + 0.629i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.960527 - 2.71374i\)
\(L(\frac12)\) \(\approx\) \(0.960527 - 2.71374i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
7 \( 1 + (0.901 + 2.48i)T \)
good2 \( 1 + (-1.68 + 1.68i)T - 2iT^{2} \)
11 \( 1 - 1.54T + 11T^{2} \)
13 \( 1 + (4.16 - 4.16i)T - 13iT^{2} \)
17 \( 1 + (-2.05 - 2.05i)T + 17iT^{2} \)
19 \( 1 - 5.70T + 19T^{2} \)
23 \( 1 + (3.72 + 3.72i)T + 23iT^{2} \)
29 \( 1 - 9.79iT - 29T^{2} \)
31 \( 1 + 3.81iT - 31T^{2} \)
37 \( 1 + (-2.81 + 2.81i)T - 37iT^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 + (-1.02 - 1.02i)T + 43iT^{2} \)
47 \( 1 + (2.05 + 2.05i)T + 47iT^{2} \)
53 \( 1 + (-4.89 - 4.89i)T + 53iT^{2} \)
59 \( 1 - 7.07T + 59T^{2} \)
61 \( 1 - 15.2iT - 61T^{2} \)
67 \( 1 + (5.31 - 5.31i)T - 67iT^{2} \)
71 \( 1 + 4.45T + 71T^{2} \)
73 \( 1 + (0.770 - 0.770i)T - 73iT^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 + (-9.60 + 9.60i)T - 83iT^{2} \)
89 \( 1 - 9.52T + 89T^{2} \)
97 \( 1 + (5.09 + 5.09i)T + 97iT^{2} \)
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

−10.62494077198725666246271928061, −9.941347607279126705200824120465, −9.084977366763754496831652990987, −7.56637413594771573769089139444, −6.79809764638173122312469617057, −5.56561492692767181814891932880, −4.37052218964212579109481320248, −3.67393266679190886349851333670, −2.52939447010028036211230243406, −1.29087137994479677494801858038, 2.74905430556920650976795103694, 3.58525651771602167533169743591, 4.91347487993808889766461302106, 5.49270938827351742627713259538, 6.41003389310259410104157148992, 7.63511757141207557660480637738, 8.058457984881291849001414952509, 9.463786668049746685839473479070, 9.960665673036906178435872935598, 11.72558489680396543620670087303

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