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  − 2.24·2-s + 0.297·3-s + 3.02·4-s + 1.49·5-s − 0.667·6-s + 2.00·7-s − 2.29·8-s − 2.91·9-s − 3.35·10-s − 0.727·11-s + 0.901·12-s − 6.17·13-s − 4.49·14-s + 0.446·15-s − 0.899·16-s − 17-s + 6.52·18-s − 3.49·19-s + 4.53·20-s + 0.596·21-s + 1.63·22-s − 5.48·23-s − 0.684·24-s − 2.75·25-s + 13.8·26-s − 1.76·27-s + 6.06·28-s + ⋯
L(s)  = 1  − 1.58·2-s + 0.171·3-s + 1.51·4-s + 0.669·5-s − 0.272·6-s + 0.757·7-s − 0.812·8-s − 0.970·9-s − 1.06·10-s − 0.219·11-s + 0.260·12-s − 1.71·13-s − 1.20·14-s + 0.115·15-s − 0.224·16-s − 0.242·17-s + 1.53·18-s − 0.801·19-s + 1.01·20-s + 0.130·21-s + 0.347·22-s − 1.14·23-s − 0.139·24-s − 0.551·25-s + 2.71·26-s − 0.338·27-s + 1.14·28-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 \( 1 + 2.24T + 2T^{2} \)
3 \( 1 - 0.297T + 3T^{2} \)
5 \( 1 - 1.49T + 5T^{2} \)
7 \( 1 - 2.00T + 7T^{2} \)
11 \( 1 + 0.727T + 11T^{2} \)
13 \( 1 + 6.17T + 13T^{2} \)
19 \( 1 + 3.49T + 19T^{2} \)
23 \( 1 + 5.48T + 23T^{2} \)
29 \( 1 - 7.44T + 29T^{2} \)
31 \( 1 + 4.37T + 31T^{2} \)
37 \( 1 - 4.71T + 37T^{2} \)
41 \( 1 + 7.24T + 41T^{2} \)
47 \( 1 + 0.886T + 47T^{2} \)
53 \( 1 - 6.71T + 53T^{2} \)
59 \( 1 - 7.08T + 59T^{2} \)
61 \( 1 - 1.56T + 61T^{2} \)
67 \( 1 + 9.37T + 67T^{2} \)
71 \( 1 + 1.41T + 71T^{2} \)
73 \( 1 + 2.66T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 + 2.32T + 83T^{2} \)
89 \( 1 + 6.18T + 89T^{2} \)
97 \( 1 + 1.30T + 97T^{2} \)
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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

−9.986426735464371452212739852933, −9.063420295020176110904970717093, −8.330148510885956652729408828575, −7.73649119909683128436427399903, −6.74405165922723822275738692125, −5.66088088436561406179123231277, −4.56610985216660071580919635244, −2.58519782493905735782650059119, −1.89555902249478005299047405181, 0, 1.89555902249478005299047405181, 2.58519782493905735782650059119, 4.56610985216660071580919635244, 5.66088088436561406179123231277, 6.74405165922723822275738692125, 7.73649119909683128436427399903, 8.330148510885956652729408828575, 9.063420295020176110904970717093, 9.986426735464371452212739852933

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