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  + 0.291·2-s − 0.858·3-s − 1.91·4-s + 3.99·5-s − 0.250·6-s − 2.74·7-s − 1.14·8-s − 2.26·9-s + 1.16·10-s + 3.05·11-s + 1.64·12-s − 5.80·13-s − 0.799·14-s − 3.42·15-s + 3.49·16-s − 17-s − 0.660·18-s − 3.69·19-s − 7.65·20-s + 2.35·21-s + 0.891·22-s − 2.72·23-s + 0.979·24-s + 10.9·25-s − 1.69·26-s + 4.51·27-s + 5.24·28-s + ⋯
L(s)  = 1  + 0.206·2-s − 0.495·3-s − 0.957·4-s + 1.78·5-s − 0.102·6-s − 1.03·7-s − 0.403·8-s − 0.754·9-s + 0.368·10-s + 0.921·11-s + 0.474·12-s − 1.61·13-s − 0.213·14-s − 0.885·15-s + 0.874·16-s − 0.242·17-s − 0.155·18-s − 0.846·19-s − 1.71·20-s + 0.513·21-s + 0.190·22-s − 0.567·23-s + 0.200·24-s + 2.19·25-s − 0.332·26-s + 0.869·27-s + 0.991·28-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 \( 1 - 0.291T + 2T^{2} \)
3 \( 1 + 0.858T + 3T^{2} \)
5 \( 1 - 3.99T + 5T^{2} \)
7 \( 1 + 2.74T + 7T^{2} \)
11 \( 1 - 3.05T + 11T^{2} \)
13 \( 1 + 5.80T + 13T^{2} \)
19 \( 1 + 3.69T + 19T^{2} \)
23 \( 1 + 2.72T + 23T^{2} \)
29 \( 1 + 10.1T + 29T^{2} \)
31 \( 1 - 3.83T + 31T^{2} \)
37 \( 1 + 9.47T + 37T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
47 \( 1 - 7.25T + 47T^{2} \)
53 \( 1 + 0.0256T + 53T^{2} \)
59 \( 1 + 4.19T + 59T^{2} \)
61 \( 1 + 6.73T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + 8.72T + 71T^{2} \)
73 \( 1 - 3.33T + 73T^{2} \)
79 \( 1 + 6.10T + 79T^{2} \)
83 \( 1 + 8.98T + 83T^{2} \)
89 \( 1 + 0.838T + 89T^{2} \)
97 \( 1 - 17.2T + 97T^{2} \)
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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

−9.875667219238799135340935338422, −9.275398193186775127824281676308, −8.659636669692685246759069124829, −6.97302605646451616271876910484, −6.15602406937115582585653656761, −5.56621458477830555065521841142, −4.70970347629156050612747991983, −3.30481716018561055051528022488, −2.04753551522320719886945198100, 0, 2.04753551522320719886945198100, 3.30481716018561055051528022488, 4.70970347629156050612747991983, 5.56621458477830555065521841142, 6.15602406937115582585653656761, 6.97302605646451616271876910484, 8.659636669692685246759069124829, 9.275398193186775127824281676308, 9.875667219238799135340935338422

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