Properties

Degree 2
Conductor $ 3 \cdot 11 \cdot 61 $
Sign $0.794 + 0.606i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (1.30 − 0.951i)2-s + (0.309 + 0.951i)3-s + (0.500 − 1.53i)4-s + (1.30 + 0.951i)6-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (0.309 − 0.951i)11-s + 1.61·12-s + (1.30 − 0.951i)13-s + (−0.5 − 0.363i)17-s + (−0.499 + 1.53i)18-s + (0.190 + 0.587i)19-s + (−0.499 − 1.53i)22-s − 1.61·23-s + (0.809 − 0.587i)24-s + (0.309 + 0.951i)25-s + ⋯
L(s)  = 1  + (1.30 − 0.951i)2-s + (0.309 + 0.951i)3-s + (0.500 − 1.53i)4-s + (1.30 + 0.951i)6-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (0.309 − 0.951i)11-s + 1.61·12-s + (1.30 − 0.951i)13-s + (−0.5 − 0.363i)17-s + (−0.499 + 1.53i)18-s + (0.190 + 0.587i)19-s + (−0.499 − 1.53i)22-s − 1.61·23-s + (0.809 − 0.587i)24-s + (0.309 + 0.951i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2013\)    =    \(3 \cdot 11 \cdot 61\)
\( \varepsilon \)  =  $0.794 + 0.606i$
motivic weight  =  \(0\)
character  :  $\chi_{2013} (1829, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2013,\ (\ :0),\ 0.794 + 0.606i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(2.595023628\)
\(L(\frac12)\)  \(\approx\)  \(2.595023628\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11,\;61\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;11,\;61\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
5 \( 1 + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + 1.61T + T^{2} \)
29 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.441747903614451860815079359378, −8.591145763642572841227068996806, −7.965353268169358114555959370119, −6.43063132910694996140041416001, −5.59971976181835431385930481035, −5.19702476317934490884801871754, −3.95289697488769425724936961715, −3.59221123046221468534299891148, −2.87667747755197645752423453914, −1.57544670175381383041692720923, 1.68886661510674863768926731397, 2.75485978529463267966960097434, 4.11513672882863889336920846109, 4.29358346710920206862928083509, 5.81820661742036333616827116408, 6.20159755150888266028503701119, 6.88230029723289910654450813392, 7.56393890845154947448000449248, 8.326935421552818523811158676441, 9.081414514982793929138229788939

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