Properties

Degree 2
Conductor $ 11 \cdot 17 \cdot 43 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.67·2-s + 2.62·3-s + 5.13·4-s + 0.949·5-s − 7.00·6-s − 4.03·7-s − 8.37·8-s + 3.86·9-s − 2.53·10-s − 11-s + 13.4·12-s + 0.167·13-s + 10.7·14-s + 2.48·15-s + 12.1·16-s + 17-s − 10.3·18-s − 8.43·19-s + 4.87·20-s − 10.5·21-s + 2.67·22-s + 5.22·23-s − 21.9·24-s − 4.09·25-s − 0.448·26-s + 2.27·27-s − 20.7·28-s + ⋯
L(s)  = 1  − 1.88·2-s + 1.51·3-s + 2.56·4-s + 0.424·5-s − 2.85·6-s − 1.52·7-s − 2.96·8-s + 1.28·9-s − 0.801·10-s − 0.301·11-s + 3.88·12-s + 0.0465·13-s + 2.88·14-s + 0.642·15-s + 3.02·16-s + 0.242·17-s − 2.43·18-s − 1.93·19-s + 1.09·20-s − 2.30·21-s + 0.569·22-s + 1.09·23-s − 4.48·24-s − 0.819·25-s − 0.0878·26-s + 0.437·27-s − 3.91·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8041\)    =    \(11 \cdot 17 \cdot 43\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8041} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8041,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{11,\;17,\;43\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{11,\;17,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad11 \( 1 + T \)
17 \( 1 - T \)
43 \( 1 - T \)
good2 \( 1 + 2.67T + 2T^{2} \)
3 \( 1 - 2.62T + 3T^{2} \)
5 \( 1 - 0.949T + 5T^{2} \)
7 \( 1 + 4.03T + 7T^{2} \)
13 \( 1 - 0.167T + 13T^{2} \)
19 \( 1 + 8.43T + 19T^{2} \)
23 \( 1 - 5.22T + 23T^{2} \)
29 \( 1 - 8.85T + 29T^{2} \)
31 \( 1 - 5.94T + 31T^{2} \)
37 \( 1 - 5.66T + 37T^{2} \)
41 \( 1 + 1.71T + 41T^{2} \)
47 \( 1 - 2.40T + 47T^{2} \)
53 \( 1 + 7.83T + 53T^{2} \)
59 \( 1 + 9.07T + 59T^{2} \)
61 \( 1 - 6.13T + 61T^{2} \)
67 \( 1 + 4.24T + 67T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 - 4.99T + 73T^{2} \)
79 \( 1 - 4.03T + 79T^{2} \)
83 \( 1 + 7.69T + 83T^{2} \)
89 \( 1 - 3.38T + 89T^{2} \)
97 \( 1 + 6.26T + 97T^{2} \)
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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

−7.86670947373231378672419101183, −6.99402787799156970713914310678, −6.46997602991848481113044508114, −5.98860633270032463695736057437, −4.39953008098659833466464187872, −3.30486234934413320526843003809, −2.72207523205226167892402742792, −2.30133425189935033148140295977, −1.22497251853944738797989176570, 0, 1.22497251853944738797989176570, 2.30133425189935033148140295977, 2.72207523205226167892402742792, 3.30486234934413320526843003809, 4.39953008098659833466464187872, 5.98860633270032463695736057437, 6.46997602991848481113044508114, 6.99402787799156970713914310678, 7.86670947373231378672419101183

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