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  − 0.231·2-s − 1.94·4-s − 3.70·5-s − 0.911·7-s + 0.912·8-s + 0.857·10-s + 11-s − 1.28·13-s + 0.210·14-s + 3.68·16-s + 7.36·17-s + 5.91·19-s + 7.21·20-s − 0.231·22-s − 6.40·23-s + 8.74·25-s + 0.298·26-s + 1.77·28-s − 9.58·29-s − 8.65·31-s − 2.67·32-s − 1.70·34-s + 3.37·35-s − 8.74·37-s − 1.36·38-s − 3.38·40-s − 6.54·41-s + ⋯
L(s)  = 1  − 0.163·2-s − 0.973·4-s − 1.65·5-s − 0.344·7-s + 0.322·8-s + 0.271·10-s + 0.301·11-s − 0.357·13-s + 0.0563·14-s + 0.920·16-s + 1.78·17-s + 1.35·19-s + 1.61·20-s − 0.0493·22-s − 1.33·23-s + 1.74·25-s + 0.0585·26-s + 0.335·28-s − 1.77·29-s − 1.55·31-s − 0.473·32-s − 0.292·34-s + 0.570·35-s − 1.43·37-s − 0.221·38-s − 0.534·40-s − 1.02·41-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  =  $(2,\ 6039,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4972333754$
$L(\frac12)$  $\approx$  $0.4972333754$
$L(\frac{3}{2})$   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 \( 1 + 0.231T + 2T^{2} \)
5 \( 1 + 3.70T + 5T^{2} \)
7 \( 1 + 0.911T + 7T^{2} \)
13 \( 1 + 1.28T + 13T^{2} \)
17 \( 1 - 7.36T + 17T^{2} \)
19 \( 1 - 5.91T + 19T^{2} \)
23 \( 1 + 6.40T + 23T^{2} \)
29 \( 1 + 9.58T + 29T^{2} \)
31 \( 1 + 8.65T + 31T^{2} \)
37 \( 1 + 8.74T + 37T^{2} \)
41 \( 1 + 6.54T + 41T^{2} \)
43 \( 1 - 9.59T + 43T^{2} \)
47 \( 1 + 5.82T + 47T^{2} \)
53 \( 1 - 5.01T + 53T^{2} \)
59 \( 1 + 1.55T + 59T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 - 0.506T + 71T^{2} \)
73 \( 1 + 3.71T + 73T^{2} \)
79 \( 1 + 5.28T + 79T^{2} \)
83 \( 1 + 7.43T + 83T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 - 19.0T + 97T^{2} \)
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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

−7.924596374062747487185167584371, −7.58866600601016836375102969611, −7.01924054918251773804005510841, −5.60750972412211380648423368213, −5.30377027750610365184036307740, −4.22992755434547474909443997993, −3.53722078016262997921803953382, −3.37125103941500529275129872942, −1.56156717444436071913658005399, −0.39700894289407426371349317147, 0.39700894289407426371349317147, 1.56156717444436071913658005399, 3.37125103941500529275129872942, 3.53722078016262997921803953382, 4.22992755434547474909443997993, 5.30377027750610365184036307740, 5.60750972412211380648423368213, 7.01924054918251773804005510841, 7.58866600601016836375102969611, 7.924596374062747487185167584371

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