Properties

Degree 2
Conductor $ 3^{2} \cdot 11 \cdot 61 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.67·2-s + 0.795·4-s + 1.45·5-s − 0.113·7-s − 2.01·8-s + 2.44·10-s + 11-s − 2.41·13-s − 0.190·14-s − 4.95·16-s − 4.33·17-s + 5.52·19-s + 1.16·20-s + 1.67·22-s + 5.61·23-s − 2.86·25-s − 4.04·26-s − 0.0906·28-s + 2.97·29-s + 3.23·31-s − 4.26·32-s − 7.25·34-s − 0.166·35-s + 5.47·37-s + 9.24·38-s − 2.93·40-s + 3.54·41-s + ⋯
L(s)  = 1  + 1.18·2-s + 0.397·4-s + 0.652·5-s − 0.0430·7-s − 0.711·8-s + 0.771·10-s + 0.301·11-s − 0.670·13-s − 0.0509·14-s − 1.23·16-s − 1.05·17-s + 1.26·19-s + 0.259·20-s + 0.356·22-s + 1.17·23-s − 0.573·25-s − 0.793·26-s − 0.0171·28-s + 0.552·29-s + 0.581·31-s − 0.753·32-s − 1.24·34-s − 0.0281·35-s + 0.900·37-s + 1.49·38-s − 0.464·40-s + 0.554·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6039} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6039,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.747677878$
$L(\frac12)$  $\approx$  $3.747677878$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11,\;61\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;61\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
11 \( 1 - T \)
61 \( 1 - T \)
good2 \( 1 - 1.67T + 2T^{2} \)
5 \( 1 - 1.45T + 5T^{2} \)
7 \( 1 + 0.113T + 7T^{2} \)
13 \( 1 + 2.41T + 13T^{2} \)
17 \( 1 + 4.33T + 17T^{2} \)
19 \( 1 - 5.52T + 19T^{2} \)
23 \( 1 - 5.61T + 23T^{2} \)
29 \( 1 - 2.97T + 29T^{2} \)
31 \( 1 - 3.23T + 31T^{2} \)
37 \( 1 - 5.47T + 37T^{2} \)
41 \( 1 - 3.54T + 41T^{2} \)
43 \( 1 - 3.64T + 43T^{2} \)
47 \( 1 - 6.81T + 47T^{2} \)
53 \( 1 - 0.123T + 53T^{2} \)
59 \( 1 - 14.8T + 59T^{2} \)
67 \( 1 - 9.16T + 67T^{2} \)
71 \( 1 - 9.69T + 71T^{2} \)
73 \( 1 + 1.05T + 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 - 2.01T + 83T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 + 4.43T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.004635683792721137245867854146, −7.02092520641100181588265698062, −6.55313351107885725593034692754, −5.72639199233544018766250380374, −5.18445126302614715099943882889, −4.52073052460563467982330655852, −3.78276155701650674549205591884, −2.82266899163281572795023265365, −2.27155712664817350507168110389, −0.849488502495908868069502440206, 0.849488502495908868069502440206, 2.27155712664817350507168110389, 2.82266899163281572795023265365, 3.78276155701650674549205591884, 4.52073052460563467982330655852, 5.18445126302614715099943882889, 5.72639199233544018766250380374, 6.55313351107885725593034692754, 7.02092520641100181588265698062, 8.004635683792721137245867854146

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