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  + 1.61·2-s + 0.604·4-s + 3.65·5-s + 0.985·7-s − 2.25·8-s + 5.90·10-s + 11-s + 2.65·13-s + 1.59·14-s − 4.84·16-s + 6.25·17-s + 1.35·19-s + 2.21·20-s + 1.61·22-s − 2.24·23-s + 8.37·25-s + 4.28·26-s + 0.596·28-s + 4.46·29-s − 4.00·31-s − 3.31·32-s + 10.0·34-s + 3.60·35-s − 1.24·37-s + 2.18·38-s − 8.23·40-s − 6.27·41-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.302·4-s + 1.63·5-s + 0.372·7-s − 0.796·8-s + 1.86·10-s + 0.301·11-s + 0.735·13-s + 0.425·14-s − 1.21·16-s + 1.51·17-s + 0.310·19-s + 0.494·20-s + 0.344·22-s − 0.468·23-s + 1.67·25-s + 0.839·26-s + 0.112·28-s + 0.830·29-s − 0.719·31-s − 0.585·32-s + 1.73·34-s + 0.609·35-s − 0.204·37-s + 0.354·38-s − 1.30·40-s − 0.980·41-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  =  $(2,\ 6039,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.354959946$
$L(\frac12)$  $\approx$  $5.354959946$
$L(\frac{3}{2})$   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 \( 1 - 1.61T + 2T^{2} \)
5 \( 1 - 3.65T + 5T^{2} \)
7 \( 1 - 0.985T + 7T^{2} \)
13 \( 1 - 2.65T + 13T^{2} \)
17 \( 1 - 6.25T + 17T^{2} \)
19 \( 1 - 1.35T + 19T^{2} \)
23 \( 1 + 2.24T + 23T^{2} \)
29 \( 1 - 4.46T + 29T^{2} \)
31 \( 1 + 4.00T + 31T^{2} \)
37 \( 1 + 1.24T + 37T^{2} \)
41 \( 1 + 6.27T + 41T^{2} \)
43 \( 1 - 0.305T + 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 + 8.28T + 59T^{2} \)
67 \( 1 + 15.5T + 67T^{2} \)
71 \( 1 - 8.28T + 71T^{2} \)
73 \( 1 - 8.53T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 + 4.84T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−8.100253474841690719683343885583, −7.08002344181034158542310795932, −6.28436861038005779311668786686, −5.74477176539925143906612745510, −5.36797731659165815257050647655, −4.57711904967819419792527716108, −3.64303739216328623742724847192, −2.96345627100972562745958370557, −1.99008670884630616827382981468, −1.11791820589654811379857328239, 1.11791820589654811379857328239, 1.99008670884630616827382981468, 2.96345627100972562745958370557, 3.64303739216328623742724847192, 4.57711904967819419792527716108, 5.36797731659165815257050647655, 5.74477176539925143906612745510, 6.28436861038005779311668786686, 7.08002344181034158542310795932, 8.100253474841690719683343885583

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