Properties

Degree 2
Conductor $ 11 \cdot 547 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.70·2-s + 2.96·3-s + 5.33·4-s − 1.26·5-s − 8.03·6-s + 1.85·7-s − 9.03·8-s + 5.79·9-s + 3.41·10-s + 11-s + 15.8·12-s − 5.34·13-s − 5.01·14-s − 3.73·15-s + 13.7·16-s + 3.21·17-s − 15.6·18-s − 7.60·19-s − 6.72·20-s + 5.48·21-s − 2.70·22-s − 3.07·23-s − 26.7·24-s − 3.40·25-s + 14.4·26-s + 8.28·27-s + 9.87·28-s + ⋯
L(s)  = 1  − 1.91·2-s + 1.71·3-s + 2.66·4-s − 0.563·5-s − 3.27·6-s + 0.699·7-s − 3.19·8-s + 1.93·9-s + 1.08·10-s + 0.301·11-s + 4.56·12-s − 1.48·13-s − 1.33·14-s − 0.965·15-s + 3.44·16-s + 0.778·17-s − 3.69·18-s − 1.74·19-s − 1.50·20-s + 1.19·21-s − 0.577·22-s − 0.642·23-s − 5.46·24-s − 0.681·25-s + 2.84·26-s + 1.59·27-s + 1.86·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6017\)    =    \(11 \cdot 547\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6017} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6017,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{11,\;547\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{11,\;547\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad11 \( 1 - T \)
547 \( 1 - T \)
good2 \( 1 + 2.70T + 2T^{2} \)
3 \( 1 - 2.96T + 3T^{2} \)
5 \( 1 + 1.26T + 5T^{2} \)
7 \( 1 - 1.85T + 7T^{2} \)
13 \( 1 + 5.34T + 13T^{2} \)
17 \( 1 - 3.21T + 17T^{2} \)
19 \( 1 + 7.60T + 19T^{2} \)
23 \( 1 + 3.07T + 23T^{2} \)
29 \( 1 + 0.603T + 29T^{2} \)
31 \( 1 - 3.99T + 31T^{2} \)
37 \( 1 + 0.667T + 37T^{2} \)
41 \( 1 - 0.907T + 41T^{2} \)
43 \( 1 + 7.50T + 43T^{2} \)
47 \( 1 - 2.39T + 47T^{2} \)
53 \( 1 - 11.9T + 53T^{2} \)
59 \( 1 - 2.90T + 59T^{2} \)
61 \( 1 - 0.749T + 61T^{2} \)
67 \( 1 + 6.13T + 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 - 2.46T + 73T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 + 7.98T + 83T^{2} \)
89 \( 1 - 3.27T + 89T^{2} \)
97 \( 1 + 4.16T + 97T^{2} \)
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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

−8.090814941676921086523385384337, −7.35562800962988418998504565068, −7.03459925318599316418072309627, −5.95300892840811652165828647929, −4.55942509633949279978318965659, −3.72146127287962546104596487355, −2.71594406531401070151271877608, −2.17351296516141217494438967283, −1.45691628661973940319850265563, 0, 1.45691628661973940319850265563, 2.17351296516141217494438967283, 2.71594406531401070151271877608, 3.72146127287962546104596487355, 4.55942509633949279978318965659, 5.95300892840811652165828647929, 7.03459925318599316418072309627, 7.35562800962988418998504565068, 8.090814941676921086523385384337

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