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  + 0.546·2-s − 1.70·4-s + 0.842·5-s − 4.19·7-s − 2.02·8-s + 0.460·10-s + 11-s − 5.95·13-s − 2.29·14-s + 2.29·16-s − 0.386·17-s − 2.95·19-s − 1.43·20-s + 0.546·22-s + 0.974·23-s − 4.28·25-s − 3.25·26-s + 7.13·28-s + 10.0·29-s − 10.6·31-s + 5.29·32-s − 0.210·34-s − 3.53·35-s − 3.16·37-s − 1.61·38-s − 1.70·40-s + 1.29·41-s + ⋯
L(s)  = 1  + 0.386·2-s − 0.850·4-s + 0.376·5-s − 1.58·7-s − 0.714·8-s + 0.145·10-s + 0.301·11-s − 1.65·13-s − 0.612·14-s + 0.574·16-s − 0.0936·17-s − 0.677·19-s − 0.320·20-s + 0.116·22-s + 0.203·23-s − 0.857·25-s − 0.638·26-s + 1.34·28-s + 1.86·29-s − 1.90·31-s + 0.936·32-s − 0.0361·34-s − 0.597·35-s − 0.520·37-s − 0.261·38-s − 0.269·40-s + 0.201·41-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  =  $(2,\ 6039,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6401652982$
$L(\frac12)$  $\approx$  $0.6401652982$
$L(\frac{3}{2})$   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 \( 1 - 0.546T + 2T^{2} \)
5 \( 1 - 0.842T + 5T^{2} \)
7 \( 1 + 4.19T + 7T^{2} \)
13 \( 1 + 5.95T + 13T^{2} \)
17 \( 1 + 0.386T + 17T^{2} \)
19 \( 1 + 2.95T + 19T^{2} \)
23 \( 1 - 0.974T + 23T^{2} \)
29 \( 1 - 10.0T + 29T^{2} \)
31 \( 1 + 10.6T + 31T^{2} \)
37 \( 1 + 3.16T + 37T^{2} \)
41 \( 1 - 1.29T + 41T^{2} \)
43 \( 1 + 4.99T + 43T^{2} \)
47 \( 1 + 7.87T + 47T^{2} \)
53 \( 1 + 0.370T + 53T^{2} \)
59 \( 1 + 13.7T + 59T^{2} \)
67 \( 1 - 9.80T + 67T^{2} \)
71 \( 1 - 4.36T + 71T^{2} \)
73 \( 1 + 2.06T + 73T^{2} \)
79 \( 1 + 8.67T + 79T^{2} \)
83 \( 1 - 7.09T + 83T^{2} \)
89 \( 1 - 7.42T + 89T^{2} \)
97 \( 1 + 6.00T + 97T^{2} \)
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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

−8.123848212917226880179744356624, −7.20577204785012116391738726783, −6.53083997089555379651916453145, −5.96604721114185222324543238324, −5.11646612116491789906685287017, −4.51662995792797156454886431476, −3.58080779383165498812684209924, −3.01103865633455867930619412372, −2.02581154554017747141564947613, −0.37441049610034581639960265431, 0.37441049610034581639960265431, 2.02581154554017747141564947613, 3.01103865633455867930619412372, 3.58080779383165498812684209924, 4.51662995792797156454886431476, 5.11646612116491789906685287017, 5.96604721114185222324543238324, 6.53083997089555379651916453145, 7.20577204785012116391738726783, 8.123848212917226880179744356624

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