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  + 1.93·2-s + 1.76·4-s − 3.07·5-s − 3.15·7-s − 0.464·8-s − 5.96·10-s + 11-s − 4.31·13-s − 6.12·14-s − 4.42·16-s + 3.81·17-s − 4.23·19-s − 5.41·20-s + 1.93·22-s − 6.48·23-s + 4.47·25-s − 8.35·26-s − 5.56·28-s + 3.27·29-s + 10.0·31-s − 7.64·32-s + 7.39·34-s + 9.72·35-s − 1.85·37-s − 8.20·38-s + 1.42·40-s + 10.7·41-s + ⋯
L(s)  = 1  + 1.37·2-s + 0.880·4-s − 1.37·5-s − 1.19·7-s − 0.164·8-s − 1.88·10-s + 0.301·11-s − 1.19·13-s − 1.63·14-s − 1.10·16-s + 0.924·17-s − 0.971·19-s − 1.21·20-s + 0.413·22-s − 1.35·23-s + 0.894·25-s − 1.63·26-s − 1.05·28-s + 0.608·29-s + 1.81·31-s − 1.35·32-s + 1.26·34-s + 1.64·35-s − 0.304·37-s − 1.33·38-s + 0.225·40-s + 1.68·41-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  =  $(2,\ 6039,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.496536420$
$L(\frac12)$  $\approx$  $1.496536420$
$L(\frac{3}{2})$   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 \( 1 - 1.93T + 2T^{2} \)
5 \( 1 + 3.07T + 5T^{2} \)
7 \( 1 + 3.15T + 7T^{2} \)
13 \( 1 + 4.31T + 13T^{2} \)
17 \( 1 - 3.81T + 17T^{2} \)
19 \( 1 + 4.23T + 19T^{2} \)
23 \( 1 + 6.48T + 23T^{2} \)
29 \( 1 - 3.27T + 29T^{2} \)
31 \( 1 - 10.0T + 31T^{2} \)
37 \( 1 + 1.85T + 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 + 8.28T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
67 \( 1 + 5.80T + 67T^{2} \)
71 \( 1 - 3.40T + 71T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 - 4.26T + 79T^{2} \)
83 \( 1 + 17.4T + 83T^{2} \)
89 \( 1 - 9.75T + 89T^{2} \)
97 \( 1 - 0.814T + 97T^{2} \)
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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

−7.899540835485542156489957952735, −7.22541058651136139982653761446, −6.39154794750972841079343456548, −6.04366857917153423150645402804, −4.94284445819420673185669348550, −4.31311770269378356021090774977, −3.81429272461206609087985447057, −3.06886779591080548569233090515, −2.41435986751857646079742662650, −0.48811443910286370109187285428, 0.48811443910286370109187285428, 2.41435986751857646079742662650, 3.06886779591080548569233090515, 3.81429272461206609087985447057, 4.31311770269378356021090774977, 4.94284445819420673185669348550, 6.04366857917153423150645402804, 6.39154794750972841079343456548, 7.22541058651136139982653761446, 7.899540835485542156489957952735

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