Properties

Degree 2
Conductor $ 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  − 2.35·2-s + 1.68·3-s + 3.56·4-s + 0.252·5-s − 3.96·6-s − 3.41·7-s − 3.68·8-s − 0.170·9-s − 0.594·10-s + 0.120·11-s + 5.99·12-s + 3.34·13-s + 8.04·14-s + 0.424·15-s + 1.56·16-s − 17-s + 0.401·18-s − 8.40·19-s + 0.897·20-s − 5.73·21-s − 0.284·22-s + 0.438·23-s − 6.19·24-s − 4.93·25-s − 7.89·26-s − 5.33·27-s − 12.1·28-s + ⋯
L(s)  = 1  − 1.66·2-s + 0.971·3-s + 1.78·4-s + 0.112·5-s − 1.61·6-s − 1.28·7-s − 1.30·8-s − 0.0568·9-s − 0.187·10-s + 0.0364·11-s + 1.72·12-s + 0.928·13-s + 2.15·14-s + 0.109·15-s + 0.390·16-s − 0.242·17-s + 0.0947·18-s − 1.92·19-s + 0.200·20-s − 1.25·21-s − 0.0607·22-s + 0.0913·23-s − 1.26·24-s − 0.987·25-s − 1.54·26-s − 1.02·27-s − 2.29·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(731\)    =    \(17 \cdot 43\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{731} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 731,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{17,\;43\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{17,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad17 \( 1 + T \)
43 \( 1 + T \)
good2 \( 1 + 2.35T + 2T^{2} \)
3 \( 1 - 1.68T + 3T^{2} \)
5 \( 1 - 0.252T + 5T^{2} \)
7 \( 1 + 3.41T + 7T^{2} \)
11 \( 1 - 0.120T + 11T^{2} \)
13 \( 1 - 3.34T + 13T^{2} \)
19 \( 1 + 8.40T + 19T^{2} \)
23 \( 1 - 0.438T + 23T^{2} \)
29 \( 1 + 4.34T + 29T^{2} \)
31 \( 1 - 7.74T + 31T^{2} \)
37 \( 1 + 8.36T + 37T^{2} \)
41 \( 1 - 5.90T + 41T^{2} \)
47 \( 1 - 1.29T + 47T^{2} \)
53 \( 1 + 4.96T + 53T^{2} \)
59 \( 1 + 4.88T + 59T^{2} \)
61 \( 1 + 12.9T + 61T^{2} \)
67 \( 1 - 6.34T + 67T^{2} \)
71 \( 1 - 11.5T + 71T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 - 8.52T + 89T^{2} \)
97 \( 1 + 6.27T + 97T^{2} \)
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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

−9.664544798503166196931334231924, −9.060407446914816188072600948173, −8.481508533506207540359404371932, −7.76972990172683476132838094252, −6.63102518749781085995156965377, −6.08689915276856061145258482038, −3.99543869560287126613495169358, −2.89263121935266338762566114469, −1.87901819194857645507979719686, 0, 1.87901819194857645507979719686, 2.89263121935266338762566114469, 3.99543869560287126613495169358, 6.08689915276856061145258482038, 6.63102518749781085995156965377, 7.76972990172683476132838094252, 8.481508533506207540359404371932, 9.060407446914816188072600948173, 9.664544798503166196931334231924

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