↑

Properties

Label 2-525-35.4-c1-0-0
Degree $2$
Conductor $525$
Sign $-0.924 - 0.381i$
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  + (−1.22 + 0.707i)2-s + (−0.866 − 0.5i)3-s + 1.41·6-s + (−0.358 − 2.62i)7-s − 2.82i·8-s + (0.499 + 0.866i)9-s + (−0.292 + 0.507i)11-s + 4.41i·13-s + (2.29 + 2.95i)14-s + (2.00 + 3.46i)16-s + (−1.94 − 1.12i)17-s + (−1.22 − 0.707i)18-s + (2.32 + 4.03i)19-s + (−1 + 2.44i)21-s − 0.828i·22-s + (−1.94 + 1.12i)23-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)2-s + (−0.499 − 0.288i)3-s + 0.577·6-s + (−0.135 − 0.990i)7-s − 0.999i·8-s + (0.166 + 0.288i)9-s + (−0.0883 + 0.152i)11-s + 1.22i·13-s + (0.612 + 0.790i)14-s + (0.500 + 0.866i)16-s + (−0.471 − 0.271i)17-s + (−0.288 − 0.166i)18-s + (0.534 + 0.925i)19-s + (−0.218 + 0.534i)21-s − 0.176i·22-s + (−0.404 + 0.233i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.924 - 0.381i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.924 - 0.381i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0490909 + 0.247883i\)
\(L(\frac12)\) \(\approx\) \(0.0490909 + 0.247883i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
7 \( 1 + (0.358 + 2.62i)T \)
good2 \( 1 + (1.22 - 0.707i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (0.292 - 0.507i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.41iT - 13T^{2} \)
17 \( 1 + (1.94 + 1.12i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.32 - 4.03i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.94 - 1.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.24T + 29T^{2} \)
31 \( 1 + (-2.91 + 5.04i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.28 - 4.20i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 - 7.58iT - 43T^{2} \)
47 \( 1 + (11.5 - 6.65i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.91 - 3.41i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.707 - 1.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.24 - 3.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.8 - 6.86i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.585T + 71T^{2} \)
73 \( 1 + (10.4 + 6.03i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.32 - 5.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.58iT - 83T^{2} \)
89 \( 1 + (6.12 + 10.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.17iT - 97T^{2} \)
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

−11.20907893188050188177878354603, −10.03213964065998905172147459952, −9.597467979727521830118311009777, −8.447842540062985171422026822996, −7.54479015464634182193281668378, −6.96181085535776482497641935442, −6.10428570884220031405778058511, −4.59476423687140196445975966551, −3.64052209644467768694559724056, −1.50904583982887004958332081762, 0.21536147459206005822902078891, 2.01631098427643336000477567472, 3.33084064444120396491154224671, 5.14393297473652135828065670975, 5.53211788850614572474442447941, 6.83400589388014162397230000368, 8.206322684938568305780123857480, 8.825286290663899230999346116468, 9.695545249140147293303903205067, 10.41657752328043963709982794741

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