Properties

Label 2-3e2-3.2-c4-0-0
Degree $2$
Conductor $9$
Sign $0.577 - 0.816i$
Analytic cond. $0.930329$
Root an. cond. $0.964535$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4.24i·2-s − 1.99·4-s − 29.6i·5-s − 28·7-s + 59.3i·8-s + 125.·10-s + 16.9i·11-s − 112·13-s − 118. i·14-s − 284·16-s − 89.0i·17-s + 560·19-s + 59.3i·20-s − 71.9·22-s + 797. i·23-s + ⋯
L(s)  = 1  + 1.06i·2-s − 0.124·4-s − 1.18i·5-s − 0.571·7-s + 0.928i·8-s + 1.25·10-s + 0.140i·11-s − 0.662·13-s − 0.606i·14-s − 1.10·16-s − 0.308i·17-s + 1.55·19-s + 0.148i·20-s − 0.148·22-s + 1.50i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(0.930329\)
Root analytic conductor: \(0.964535\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{9} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 9,\ (\ :2),\ 0.577 - 0.816i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.907074 + 0.469536i\)
\(L(\frac12)\) \(\approx\) \(0.907074 + 0.469536i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 - 4.24iT - 16T^{2} \)
5 \( 1 + 29.6iT - 625T^{2} \)
7 \( 1 + 28T + 2.40e3T^{2} \)
11 \( 1 - 16.9iT - 1.46e4T^{2} \)
13 \( 1 + 112T + 2.85e4T^{2} \)
17 \( 1 + 89.0iT - 8.35e4T^{2} \)
19 \( 1 - 560T + 1.30e5T^{2} \)
23 \( 1 - 797. iT - 2.79e5T^{2} \)
29 \( 1 + 988. iT - 7.07e5T^{2} \)
31 \( 1 + 364T + 9.23e5T^{2} \)
37 \( 1 + 826T + 1.87e6T^{2} \)
41 \( 1 + 1.81e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.73e3T + 3.41e6T^{2} \)
47 \( 1 + 1.30e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.79e3iT - 7.89e6T^{2} \)
59 \( 1 - 4.51e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.61e3T + 1.38e7T^{2} \)
67 \( 1 + 3.78e3T + 2.01e7T^{2} \)
71 \( 1 + 8.60e3iT - 2.54e7T^{2} \)
73 \( 1 - 6.60e3T + 2.83e7T^{2} \)
79 \( 1 + 4.27e3T + 3.89e7T^{2} \)
83 \( 1 - 118. iT - 4.74e7T^{2} \)
89 \( 1 - 4.36e3iT - 6.27e7T^{2} \)
97 \( 1 + 5.82e3T + 8.85e7T^{2} \)
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

−20.79834550994884979452667231040, −19.68945706363571796124802158745, −17.56178094444797750077449670346, −16.45086328334429934111289160151, −15.51072000936153803521639759887, −13.69219155187576034968971006764, −11.97386217800563039958632606030, −9.324655210416613825846300216378, −7.50214236652905414179581649637, −5.39176983631017012344750806467, 3.02878905212755916404233430378, 6.87871834439669389065105973666, 9.880286965419803764732677726567, 11.08512464056210486461730968623, 12.58650245369950066745516405039, 14.43674555261655475358626398397, 16.11414509296955556598322522095, 18.24221438119862436751232998949, 19.23754369912055127718612017883, 20.39814851515352733791280225936

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