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  + 2.45·2-s + 4.04·4-s + 1.85·5-s + 2.32·7-s + 5.03·8-s + 4.56·10-s + 11-s + 6.54·13-s + 5.71·14-s + 4.28·16-s − 2.94·17-s + 6.42·19-s + 7.50·20-s + 2.45·22-s + 1.80·23-s − 1.56·25-s + 16.0·26-s + 9.40·28-s − 8.87·29-s + 2.72·31-s + 0.469·32-s − 7.25·34-s + 4.30·35-s − 3.53·37-s + 15.8·38-s + 9.33·40-s + 2.27·41-s + ⋯
L(s)  = 1  + 1.73·2-s + 2.02·4-s + 0.829·5-s + 0.878·7-s + 1.77·8-s + 1.44·10-s + 0.301·11-s + 1.81·13-s + 1.52·14-s + 1.07·16-s − 0.715·17-s + 1.47·19-s + 1.67·20-s + 0.524·22-s + 0.375·23-s − 0.312·25-s + 3.15·26-s + 1.77·28-s − 1.64·29-s + 0.488·31-s + 0.0829·32-s − 1.24·34-s + 0.728·35-s − 0.581·37-s + 2.56·38-s + 1.47·40-s + 0.355·41-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  =  $(2,\ 6039,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $8.646735178$
$L(\frac12)$  $\approx$  $8.646735178$
$L(\frac{3}{2})$   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 \( 1 - 2.45T + 2T^{2} \)
5 \( 1 - 1.85T + 5T^{2} \)
7 \( 1 - 2.32T + 7T^{2} \)
13 \( 1 - 6.54T + 13T^{2} \)
17 \( 1 + 2.94T + 17T^{2} \)
19 \( 1 - 6.42T + 19T^{2} \)
23 \( 1 - 1.80T + 23T^{2} \)
29 \( 1 + 8.87T + 29T^{2} \)
31 \( 1 - 2.72T + 31T^{2} \)
37 \( 1 + 3.53T + 37T^{2} \)
41 \( 1 - 2.27T + 41T^{2} \)
43 \( 1 + 3.10T + 43T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 + 7.91T + 53T^{2} \)
59 \( 1 - 5.84T + 59T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 - 3.60T + 79T^{2} \)
83 \( 1 - 5.22T + 83T^{2} \)
89 \( 1 - 14.1T + 89T^{2} \)
97 \( 1 - 17.9T + 97T^{2} \)
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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

−7.83104467799374384692753345809, −7.10247939953318461262193294446, −6.20353256446652803250674272276, −5.92303285546322875397459927399, −5.13984397210527539877405230030, −4.57394148709128905537151400913, −3.64348756873420302867649033040, −3.14452029841265645532242005991, −1.90365766311106273815355062741, −1.45344622938096362104872830700, 1.45344622938096362104872830700, 1.90365766311106273815355062741, 3.14452029841265645532242005991, 3.64348756873420302867649033040, 4.57394148709128905537151400913, 5.13984397210527539877405230030, 5.92303285546322875397459927399, 6.20353256446652803250674272276, 7.10247939953318461262193294446, 7.83104467799374384692753345809

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