↑

Properties

Label 2-15e2-5.4-c9-0-60
Degree $2$
Conductor $225$
Sign $-0.447 + 0.894i$
Analytic cond. $115.883$
Root an. cond. $10.7648$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 512·4-s − 1.25e4i·7-s − 1.18e5i·13-s + 2.62e5·16-s + 9.76e5·19-s − 6.44e6i·28-s + 1.69e6·31-s − 1.53e7i·37-s + 1.65e7i·43-s − 1.17e8·49-s − 6.06e7i·52-s − 1.17e8·61-s + 1.34e8·64-s + 1.12e8i·67-s − 2.96e8i·73-s + ⋯
L(s)  = 1  + 4-s − 1.98i·7-s − 1.14i·13-s + 16-s + 1.71·19-s − 1.98i·28-s + 0.328·31-s − 1.34i·37-s + 0.739i·43-s − 2.92·49-s − 1.14i·52-s − 1.09·61-s + 64-s + 0.682i·67-s − 1.22i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(115.883\)
Root analytic conductor: \(10.7648\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :9/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.915108256\)
\(L(\frac12)\) \(\approx\) \(2.915108256\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 - 512T^{2} \)
7 \( 1 + 1.25e4iT - 4.03e7T^{2} \)
11 \( 1 + 2.35e9T^{2} \)
13 \( 1 + 1.18e5iT - 1.06e10T^{2} \)
17 \( 1 - 1.18e11T^{2} \)
19 \( 1 - 9.76e5T + 3.22e11T^{2} \)
23 \( 1 - 1.80e12T^{2} \)
29 \( 1 + 1.45e13T^{2} \)
31 \( 1 - 1.69e6T + 2.64e13T^{2} \)
37 \( 1 + 1.53e7iT - 1.29e14T^{2} \)
41 \( 1 + 3.27e14T^{2} \)
43 \( 1 - 1.65e7iT - 5.02e14T^{2} \)
47 \( 1 - 1.11e15T^{2} \)
53 \( 1 - 3.29e15T^{2} \)
59 \( 1 + 8.66e15T^{2} \)
61 \( 1 + 1.17e8T + 1.16e16T^{2} \)
67 \( 1 - 1.12e8iT - 2.72e16T^{2} \)
71 \( 1 + 4.58e16T^{2} \)
73 \( 1 + 2.96e8iT - 5.88e16T^{2} \)
79 \( 1 - 6.16e8T + 1.19e17T^{2} \)
83 \( 1 - 1.86e17T^{2} \)
89 \( 1 + 3.50e17T^{2} \)
97 \( 1 - 1.28e9iT - 7.60e17T^{2} \)
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

−10.49562118849523901975980949873, −9.642440723908496323496935947058, −7.79410407072435374579473543013, −7.49535569512738116916240264367, −6.44958116447373303861500756546, −5.21383824914139078657083917147, −3.80469294327556295184080848880, −2.94759747570085631319529461095, −1.34485720683193743958129498993, −0.58716910928399054899566709813, 1.43326965020406336531425643267, 2.36556646607541147484191629425, 3.23848588139330863378160602969, 5.03936479167339104379147504950, 5.93319801743028770781567507383, 6.80660408008047877738005933672, 7.993606090568720206718773898082, 9.034379858999126043263678069370, 9.834008205695799409465971808116, 11.26113281845775988372388281353

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