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  − 1.18·2-s − 0.604·4-s + 2.90·5-s − 0.528·7-s + 3.07·8-s − 3.42·10-s + 11-s − 0.150·13-s + 0.624·14-s − 2.42·16-s − 0.0673·17-s − 0.482·19-s − 1.75·20-s − 1.18·22-s + 4.10·23-s + 3.41·25-s + 0.178·26-s + 0.319·28-s + 5.83·29-s − 2.84·31-s − 3.28·32-s + 0.0795·34-s − 1.53·35-s + 6.02·37-s + 0.569·38-s + 8.92·40-s − 11.4·41-s + ⋯
L(s)  = 1  − 0.835·2-s − 0.302·4-s + 1.29·5-s − 0.199·7-s + 1.08·8-s − 1.08·10-s + 0.301·11-s − 0.0418·13-s + 0.166·14-s − 0.606·16-s − 0.0163·17-s − 0.110·19-s − 0.391·20-s − 0.251·22-s + 0.856·23-s + 0.682·25-s + 0.0349·26-s + 0.0603·28-s + 1.08·29-s − 0.510·31-s − 0.581·32-s + 0.0136·34-s − 0.259·35-s + 0.991·37-s + 0.0924·38-s + 1.41·40-s − 1.78·41-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  =  $(2,\ 6039,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.473229025$
$L(\frac12)$  $\approx$  $1.473229025$
$L(\frac{3}{2})$   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 \( 1 + 1.18T + 2T^{2} \)
5 \( 1 - 2.90T + 5T^{2} \)
7 \( 1 + 0.528T + 7T^{2} \)
13 \( 1 + 0.150T + 13T^{2} \)
17 \( 1 + 0.0673T + 17T^{2} \)
19 \( 1 + 0.482T + 19T^{2} \)
23 \( 1 - 4.10T + 23T^{2} \)
29 \( 1 - 5.83T + 29T^{2} \)
31 \( 1 + 2.84T + 31T^{2} \)
37 \( 1 - 6.02T + 37T^{2} \)
41 \( 1 + 11.4T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 - 3.90T + 47T^{2} \)
53 \( 1 - 0.588T + 53T^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
67 \( 1 - 9.38T + 67T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 - 13.0T + 73T^{2} \)
79 \( 1 - 9.65T + 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + 1.66T + 97T^{2} \)
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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

−8.221721489446094244262803625259, −7.48538470267631084759690213103, −6.65627845657887208633866651495, −6.08646457178597433760199412432, −5.18461188857912481051671336487, −4.62373633007760209266167763218, −3.57500623869202196276361111457, −2.51289218475540666450587825300, −1.65322550260515605774620357453, −0.76646874793059205121306802660, 0.76646874793059205121306802660, 1.65322550260515605774620357453, 2.51289218475540666450587825300, 3.57500623869202196276361111457, 4.62373633007760209266167763218, 5.18461188857912481051671336487, 6.08646457178597433760199412432, 6.65627845657887208633866651495, 7.48538470267631084759690213103, 8.221721489446094244262803625259

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