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  − 0.0561·2-s − 1.99·4-s + 2.87·5-s + 3.53·7-s + 0.224·8-s − 0.161·10-s + 11-s + 1.66·13-s − 0.198·14-s + 3.98·16-s + 6.51·17-s − 4.27·19-s − 5.73·20-s − 0.0561·22-s + 8.02·23-s + 3.23·25-s − 0.0934·26-s − 7.06·28-s − 3.26·29-s + 6.97·31-s − 0.672·32-s − 0.365·34-s + 10.1·35-s + 0.826·37-s + 0.239·38-s + 0.644·40-s + 11.9·41-s + ⋯
L(s)  = 1  − 0.0397·2-s − 0.998·4-s + 1.28·5-s + 1.33·7-s + 0.0793·8-s − 0.0509·10-s + 0.301·11-s + 0.461·13-s − 0.0531·14-s + 0.995·16-s + 1.57·17-s − 0.979·19-s − 1.28·20-s − 0.0119·22-s + 1.67·23-s + 0.647·25-s − 0.0183·26-s − 1.33·28-s − 0.606·29-s + 1.25·31-s − 0.118·32-s − 0.0627·34-s + 1.71·35-s + 0.135·37-s + 0.0389·38-s + 0.101·40-s + 1.86·41-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  =  $(2,\ 6039,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.847070669$
$L(\frac12)$  $\approx$  $2.847070669$
$L(\frac{3}{2})$   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 \( 1 + 0.0561T + 2T^{2} \)
5 \( 1 - 2.87T + 5T^{2} \)
7 \( 1 - 3.53T + 7T^{2} \)
13 \( 1 - 1.66T + 13T^{2} \)
17 \( 1 - 6.51T + 17T^{2} \)
19 \( 1 + 4.27T + 19T^{2} \)
23 \( 1 - 8.02T + 23T^{2} \)
29 \( 1 + 3.26T + 29T^{2} \)
31 \( 1 - 6.97T + 31T^{2} \)
37 \( 1 - 0.826T + 37T^{2} \)
41 \( 1 - 11.9T + 41T^{2} \)
43 \( 1 + 2.21T + 43T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 + 8.11T + 53T^{2} \)
59 \( 1 + 2.00T + 59T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 - 2.13T + 71T^{2} \)
73 \( 1 - 2.54T + 73T^{2} \)
79 \( 1 + 6.81T + 79T^{2} \)
83 \( 1 + 0.562T + 83T^{2} \)
89 \( 1 + 7.23T + 89T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−8.197050099730088802048466013504, −7.58728014338885390873268781250, −6.47564336837382363771111181268, −5.81213088060360925104757218525, −5.13290788795829710764244355459, −4.67915976475497539596388366483, −3.74011365897333462013844152687, −2.72597386113074499817000034813, −1.57024127347974521585597910517, −1.04218039491255058316551189340, 1.04218039491255058316551189340, 1.57024127347974521585597910517, 2.72597386113074499817000034813, 3.74011365897333462013844152687, 4.67915976475497539596388366483, 5.13290788795829710764244355459, 5.81213088060360925104757218525, 6.47564336837382363771111181268, 7.58728014338885390873268781250, 8.197050099730088802048466013504

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