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  + 1.55·2-s + 0.467·3-s + 0.410·4-s − 1.37·5-s + 0.726·6-s − 3.35·7-s − 2.46·8-s − 2.78·9-s − 2.13·10-s + 2.05·11-s + 0.192·12-s − 1.60·13-s − 5.20·14-s − 0.644·15-s − 4.65·16-s − 17-s − 4.31·18-s + 4.43·19-s − 0.564·20-s − 1.56·21-s + 3.19·22-s − 2.95·23-s − 1.15·24-s − 3.10·25-s − 2.49·26-s − 2.70·27-s − 1.37·28-s + ⋯
L(s)  = 1  + 1.09·2-s + 0.270·3-s + 0.205·4-s − 0.615·5-s + 0.296·6-s − 1.26·7-s − 0.872·8-s − 0.927·9-s − 0.675·10-s + 0.620·11-s + 0.0554·12-s − 0.445·13-s − 1.39·14-s − 0.166·15-s − 1.16·16-s − 0.242·17-s − 1.01·18-s + 1.01·19-s − 0.126·20-s − 0.342·21-s + 0.681·22-s − 0.616·23-s − 0.235·24-s − 0.621·25-s − 0.488·26-s − 0.520·27-s − 0.259·28-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 \( 1 - 1.55T + 2T^{2} \)
3 \( 1 - 0.467T + 3T^{2} \)
5 \( 1 + 1.37T + 5T^{2} \)
7 \( 1 + 3.35T + 7T^{2} \)
11 \( 1 - 2.05T + 11T^{2} \)
13 \( 1 + 1.60T + 13T^{2} \)
19 \( 1 - 4.43T + 19T^{2} \)
23 \( 1 + 2.95T + 23T^{2} \)
29 \( 1 + 3.13T + 29T^{2} \)
31 \( 1 + 4.72T + 31T^{2} \)
37 \( 1 - 6.73T + 37T^{2} \)
41 \( 1 - 9.28T + 41T^{2} \)
47 \( 1 + 8.52T + 47T^{2} \)
53 \( 1 - 9.33T + 53T^{2} \)
59 \( 1 + 9.83T + 59T^{2} \)
61 \( 1 - 2.75T + 61T^{2} \)
67 \( 1 - 8.84T + 67T^{2} \)
71 \( 1 + 1.37T + 71T^{2} \)
73 \( 1 + 0.237T + 73T^{2} \)
79 \( 1 + 7.25T + 79T^{2} \)
83 \( 1 + 2.15T + 83T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 + 12.3T + 97T^{2} \)
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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

−9.636356672510607432848327470142, −9.334888056856925858937426460117, −8.213159074108972597148798098600, −7.16382076688123639464718917725, −6.14386475681703177786822373728, −5.50115242423230845791444184146, −4.17760811363505970694803797222, −3.50560523527321925559032479873, −2.65902646729624576157635952031, 0, 2.65902646729624576157635952031, 3.50560523527321925559032479873, 4.17760811363505970694803797222, 5.50115242423230845791444184146, 6.14386475681703177786822373728, 7.16382076688123639464718917725, 8.213159074108972597148798098600, 9.334888056856925858937426460117, 9.636356672510607432848327470142

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