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  + 0.179·2-s − 1.96·4-s − 2.20·5-s + 3.17·7-s − 0.713·8-s − 0.396·10-s + 11-s + 5.02·13-s + 0.570·14-s + 3.80·16-s − 1.08·17-s + 3.53·19-s + 4.33·20-s + 0.179·22-s − 1.91·23-s − 0.139·25-s + 0.903·26-s − 6.24·28-s + 5.13·29-s − 0.976·31-s + 2.11·32-s − 0.195·34-s − 6.99·35-s + 7.67·37-s + 0.636·38-s + 1.57·40-s + 2.80·41-s + ⋯
L(s)  = 1  + 0.127·2-s − 0.983·4-s − 0.986·5-s + 1.19·7-s − 0.252·8-s − 0.125·10-s + 0.301·11-s + 1.39·13-s + 0.152·14-s + 0.951·16-s − 0.263·17-s + 0.811·19-s + 0.970·20-s + 0.0383·22-s − 0.399·23-s − 0.0278·25-s + 0.177·26-s − 1.17·28-s + 0.953·29-s − 0.175·31-s + 0.373·32-s − 0.0335·34-s − 1.18·35-s + 1.26·37-s + 0.103·38-s + 0.248·40-s + 0.437·41-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  =  $(2,\ 6039,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.677198609$
$L(\frac12)$  $\approx$  $1.677198609$
$L(\frac{3}{2})$   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 \( 1 - 0.179T + 2T^{2} \)
5 \( 1 + 2.20T + 5T^{2} \)
7 \( 1 - 3.17T + 7T^{2} \)
13 \( 1 - 5.02T + 13T^{2} \)
17 \( 1 + 1.08T + 17T^{2} \)
19 \( 1 - 3.53T + 19T^{2} \)
23 \( 1 + 1.91T + 23T^{2} \)
29 \( 1 - 5.13T + 29T^{2} \)
31 \( 1 + 0.976T + 31T^{2} \)
37 \( 1 - 7.67T + 37T^{2} \)
41 \( 1 - 2.80T + 41T^{2} \)
43 \( 1 + 7.46T + 43T^{2} \)
47 \( 1 + 5.09T + 47T^{2} \)
53 \( 1 - 10.9T + 53T^{2} \)
59 \( 1 + 8.89T + 59T^{2} \)
67 \( 1 + 6.12T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 - 15.6T + 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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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

−8.007428863452165743262244827578, −7.77597428986400563260697300329, −6.63357415606150060177889304485, −5.81404060527195026311003218888, −5.03720588169550933467235291590, −4.34150806994416209431474150665, −3.87891880341058966682614856675, −3.06652319841829946492696195069, −1.59056212457827549864428479331, −0.72938973837820853887462298201, 0.72938973837820853887462298201, 1.59056212457827549864428479331, 3.06652319841829946492696195069, 3.87891880341058966682614856675, 4.34150806994416209431474150665, 5.03720588169550933467235291590, 5.81404060527195026311003218888, 6.63357415606150060177889304485, 7.77597428986400563260697300329, 8.007428863452165743262244827578

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