Properties

Degree 2
Conductor $ 11 \cdot 17 \cdot 43 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.62·2-s + 1.21·3-s + 4.91·4-s + 1.03·5-s − 3.19·6-s + 4.97·7-s − 7.66·8-s − 1.52·9-s − 2.72·10-s + 11-s + 5.96·12-s − 4.96·13-s − 13.0·14-s + 1.25·15-s + 10.3·16-s + 17-s + 4.01·18-s + 0.115·19-s + 5.09·20-s + 6.03·21-s − 2.62·22-s + 3.52·23-s − 9.30·24-s − 3.92·25-s + 13.0·26-s − 5.49·27-s + 24.4·28-s + ⋯
L(s)  = 1  − 1.85·2-s + 0.700·3-s + 2.45·4-s + 0.463·5-s − 1.30·6-s + 1.88·7-s − 2.71·8-s − 0.508·9-s − 0.861·10-s + 0.301·11-s + 1.72·12-s − 1.37·13-s − 3.49·14-s + 0.324·15-s + 2.58·16-s + 0.242·17-s + 0.946·18-s + 0.0265·19-s + 1.13·20-s + 1.31·21-s − 0.560·22-s + 0.734·23-s − 1.89·24-s − 0.785·25-s + 2.56·26-s − 1.05·27-s + 4.62·28-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  =  0
Selberg data  =  $(2,\ 8041,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.527648323$
$L(\frac12)$  $\approx$  $1.527648323$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad11 \( 1 - T \)
17 \( 1 - T \)
43 \( 1 - T \)
good2 \( 1 + 2.62T + 2T^{2} \)
3 \( 1 - 1.21T + 3T^{2} \)
5 \( 1 - 1.03T + 5T^{2} \)
7 \( 1 - 4.97T + 7T^{2} \)
13 \( 1 + 4.96T + 13T^{2} \)
19 \( 1 - 0.115T + 19T^{2} \)
23 \( 1 - 3.52T + 23T^{2} \)
29 \( 1 - 8.50T + 29T^{2} \)
31 \( 1 - 8.64T + 31T^{2} \)
37 \( 1 + 0.835T + 37T^{2} \)
41 \( 1 - 9.76T + 41T^{2} \)
47 \( 1 - 5.23T + 47T^{2} \)
53 \( 1 + 8.52T + 53T^{2} \)
59 \( 1 - 7.29T + 59T^{2} \)
61 \( 1 + 7.89T + 61T^{2} \)
67 \( 1 - 3.71T + 67T^{2} \)
71 \( 1 - 11.1T + 71T^{2} \)
73 \( 1 + 5.58T + 73T^{2} \)
79 \( 1 - 2.14T + 79T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 + 5.98T + 89T^{2} \)
97 \( 1 + 9.81T + 97T^{2} \)
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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

−7.959475547130706227913893711491, −7.61948385912995486868558126105, −6.84549136874932266046487023977, −5.96653081008715833784855849461, −5.14246780311166678814131246735, −4.32343071554495503572784825636, −2.78708694187010716726626990954, −2.45242282315895685671431973595, −1.63309479075753078277671461738, −0.812862622323073198613536583003, 0.812862622323073198613536583003, 1.63309479075753078277671461738, 2.45242282315895685671431973595, 2.78708694187010716726626990954, 4.32343071554495503572784825636, 5.14246780311166678814131246735, 5.96653081008715833784855849461, 6.84549136874932266046487023977, 7.61948385912995486868558126105, 7.959475547130706227913893711491

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