↑

Properties

Label 2-2475-2475.2012-c0-0-1
Degree $2$
Conductor $2475$
Sign $-0.361 + 0.932i$
Analytic cond. $1.23518$
Root an. cond. $1.11138$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.669 − 0.743i)3-s + (−0.207 + 0.978i)4-s + (−0.866 + 0.5i)5-s + (−0.104 + 0.994i)9-s + (−0.994 − 0.104i)11-s + (0.866 − 0.5i)12-s + (0.951 + 0.309i)15-s + (−0.913 − 0.406i)16-s + (−0.309 − 0.951i)20-s + (−0.847 − 0.325i)23-s + (0.499 − 0.866i)25-s + (0.809 − 0.587i)27-s + (−0.544 − 0.604i)31-s + (0.587 + 0.809i)33-s + (−0.951 − 0.309i)36-s + (1.90 + 0.302i)37-s + ⋯
L(s)  = 1  + (−0.669 − 0.743i)3-s + (−0.207 + 0.978i)4-s + (−0.866 + 0.5i)5-s + (−0.104 + 0.994i)9-s + (−0.994 − 0.104i)11-s + (0.866 − 0.5i)12-s + (0.951 + 0.309i)15-s + (−0.913 − 0.406i)16-s + (−0.309 − 0.951i)20-s + (−0.847 − 0.325i)23-s + (0.499 − 0.866i)25-s + (0.809 − 0.587i)27-s + (−0.544 − 0.604i)31-s + (0.587 + 0.809i)33-s + (−0.951 − 0.309i)36-s + (1.90 + 0.302i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.361 + 0.932i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (2012, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ -0.361 + 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2533951252\)
\(L(\frac12)\) \(\approx\) \(0.2533951252\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.669 + 0.743i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.994 + 0.104i)T \)
good2 \( 1 + (0.207 - 0.978i)T^{2} \)
7 \( 1 + (0.866 + 0.5i)T^{2} \)
13 \( 1 + (-0.207 - 0.978i)T^{2} \)
17 \( 1 + (0.587 - 0.809i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.847 + 0.325i)T + (0.743 + 0.669i)T^{2} \)
29 \( 1 + (0.104 + 0.994i)T^{2} \)
31 \( 1 + (0.544 + 0.604i)T + (-0.104 + 0.994i)T^{2} \)
37 \( 1 + (-1.90 - 0.302i)T + (0.951 + 0.309i)T^{2} \)
41 \( 1 + (-0.978 + 0.207i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.0270 + 0.516i)T + (-0.994 + 0.104i)T^{2} \)
53 \( 1 + (-0.847 + 1.66i)T + (-0.587 - 0.809i)T^{2} \)
59 \( 1 + (0.0218 + 0.207i)T + (-0.978 + 0.207i)T^{2} \)
61 \( 1 + (0.978 + 0.207i)T^{2} \)
67 \( 1 + (-0.104 + 1.99i)T + (-0.994 - 0.104i)T^{2} \)
71 \( 1 + (1.89 - 0.614i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.951 + 0.309i)T^{2} \)
79 \( 1 + (-0.104 - 0.994i)T^{2} \)
83 \( 1 + (-0.406 - 0.913i)T^{2} \)
89 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.715 - 0.0375i)T + (0.994 - 0.104i)T^{2} \)
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

−8.363969849053428075250780603033, −8.072015757186355477360058346269, −7.42178785822609559486869344039, −6.77006688285881323605597718017, −5.88332898939834550503193633358, −4.85721797441066034641859320634, −4.07133358044753852340146859728, −3.04119061930110333124920287122, −2.21187536239982697292496605813, −0.20242239982328667216761797896, 1.19710470300721405044675568100, 2.83581555328779682040926361751, 4.14664527590958946527518610401, 4.53920483716466628811019870172, 5.49665299440970710394095716439, 5.87999094878712285298369878403, 7.00933250433081788977217826271, 7.87218546876797165104364478757, 8.753128592724681169962615608805, 9.472198371741145676811366434689

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