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  + 2.63·2-s + 4.93·4-s − 2.62·5-s + 5.18·7-s + 7.73·8-s − 6.91·10-s + 11-s − 3.47·13-s + 13.6·14-s + 10.5·16-s − 1.94·17-s − 0.343·19-s − 12.9·20-s + 2.63·22-s + 4.48·23-s + 1.89·25-s − 9.14·26-s + 25.6·28-s + 5.47·29-s + 2.40·31-s + 12.2·32-s − 5.13·34-s − 13.6·35-s + 10.3·37-s − 0.903·38-s − 20.3·40-s + 0.436·41-s + ⋯
L(s)  = 1  + 1.86·2-s + 2.46·4-s − 1.17·5-s + 1.96·7-s + 2.73·8-s − 2.18·10-s + 0.301·11-s − 0.963·13-s + 3.65·14-s + 2.62·16-s − 0.472·17-s − 0.0787·19-s − 2.89·20-s + 0.561·22-s + 0.935·23-s + 0.378·25-s − 1.79·26-s + 4.84·28-s + 1.01·29-s + 0.432·31-s + 2.15·32-s − 0.880·34-s − 2.30·35-s + 1.70·37-s − 0.146·38-s − 3.21·40-s + 0.0681·41-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  =  $(2,\ 6039,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $7.267931177$
$L(\frac12)$  $\approx$  $7.267931177$
$L(\frac{3}{2})$   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 \( 1 - 2.63T + 2T^{2} \)
5 \( 1 + 2.62T + 5T^{2} \)
7 \( 1 - 5.18T + 7T^{2} \)
13 \( 1 + 3.47T + 13T^{2} \)
17 \( 1 + 1.94T + 17T^{2} \)
19 \( 1 + 0.343T + 19T^{2} \)
23 \( 1 - 4.48T + 23T^{2} \)
29 \( 1 - 5.47T + 29T^{2} \)
31 \( 1 - 2.40T + 31T^{2} \)
37 \( 1 - 10.3T + 37T^{2} \)
41 \( 1 - 0.436T + 41T^{2} \)
43 \( 1 - 6.40T + 43T^{2} \)
47 \( 1 + 4.42T + 47T^{2} \)
53 \( 1 + 2.47T + 53T^{2} \)
59 \( 1 + 13.7T + 59T^{2} \)
67 \( 1 - 2.76T + 67T^{2} \)
71 \( 1 - 16.0T + 71T^{2} \)
73 \( 1 + 3.11T + 73T^{2} \)
79 \( 1 + 5.15T + 79T^{2} \)
83 \( 1 + 2.12T + 83T^{2} \)
89 \( 1 + 3.14T + 89T^{2} \)
97 \( 1 - 6.51T + 97T^{2} \)
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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

−7.86323455114375661372439844208, −7.27315260468101370481981372304, −6.59636156464413398051104174795, −5.63507063350523898300299784407, −4.81533840861761459607708639583, −4.55184548599567171019139140010, −4.03050806744284436847454688038, −2.94385867771379151246537435115, −2.24681508121850018364887130734, −1.15978805539640306195740486671, 1.15978805539640306195740486671, 2.24681508121850018364887130734, 2.94385867771379151246537435115, 4.03050806744284436847454688038, 4.55184548599567171019139140010, 4.81533840861761459607708639583, 5.63507063350523898300299784407, 6.59636156464413398051104174795, 7.27315260468101370481981372304, 7.86323455114375661372439844208

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