Properties

Degree $2$
Conductor $759$
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-s + 3-s − 4-s − 2·5-s − 6-s + 3·8-s + 9-s + 2·10-s + 11-s − 12-s − 2·13-s − 2·15-s − 16-s + 2·17-s − 18-s − 4·19-s + 2·20-s − 22-s − 23-s + 3·24-s − 25-s + 2·26-s + 27-s − 2·29-s + 2·30-s − 5·32-s + 33-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.288·12-s − 0.554·13-s − 0.516·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.447·20-s − 0.213·22-s − 0.208·23-s + 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.371·29-s + 0.365·30-s − 0.883·32-s + 0.174·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 759 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 759 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(759\)    =    \(3 \cdot 11 \cdot 23\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{759} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 759,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
11 \( 1 - T \)
23 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.49296684310198, −19.17343043772184, −18.47131220094761, −17.66422679350569, −16.88380497835692, −16.32338679852934, −15.28417047194102, −14.78140069637159, −13.90405249625671, −13.19514864034125, −12.28146966933124, −11.55544357249219, −10.39991504048383, −9.895138686960886, −8.855036413037729, −8.354860151865079, −7.594084414141375, −6.801626133002743, −5.226129891998146, −4.233235805460018, −3.426952011980787, −1.767580058429338, 0, 1.767580058429338, 3.426952011980787, 4.233235805460018, 5.226129891998146, 6.801626133002743, 7.594084414141375, 8.354860151865079, 8.855036413037729, 9.895138686960886, 10.39991504048383, 11.55544357249219, 12.28146966933124, 13.19514864034125, 13.90405249625671, 14.78140069637159, 15.28417047194102, 16.32338679852934, 16.88380497835692, 17.66422679350569, 18.47131220094761, 19.17343043772184, 19.49296684310198

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