Properties

Label 2-1288-1.1-c1-0-18
Degree $2$
Conductor $1288$
Sign $-1$
Analytic cond. $10.2847$
Root an. cond. $3.20698$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 7-s + 9-s − 6·17-s + 6·19-s − 2·21-s + 23-s − 5·25-s + 4·27-s − 6·29-s + 8·31-s + 2·37-s − 2·41-s − 8·43-s − 8·47-s + 49-s + 12·51-s − 2·53-s − 12·57-s − 6·59-s + 63-s − 12·67-s − 2·69-s − 8·71-s − 6·73-s + 10·75-s − 16·79-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.45·17-s + 1.37·19-s − 0.436·21-s + 0.208·23-s − 25-s + 0.769·27-s − 1.11·29-s + 1.43·31-s + 0.328·37-s − 0.312·41-s − 1.21·43-s − 1.16·47-s + 1/7·49-s + 1.68·51-s − 0.274·53-s − 1.58·57-s − 0.781·59-s + 0.125·63-s − 1.46·67-s − 0.240·69-s − 0.949·71-s − 0.702·73-s + 1.15·75-s − 1.80·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1288\)    =    \(2^{3} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(10.2847\)
Root analytic conductor: \(3.20698\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1288,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
7 \( 1 - T \)
23 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−9.361120626920682552766967596987, −8.429630893089533244710293176268, −7.51686479596460400303178433673, −6.62689974931662635095789969685, −5.89317830408152735471088243694, −5.06992659086327375497431676892, −4.34435790835530441237828655946, −2.98835684759371857772218254259, −1.54541982304235013732105849450, 0, 1.54541982304235013732105849450, 2.98835684759371857772218254259, 4.34435790835530441237828655946, 5.06992659086327375497431676892, 5.89317830408152735471088243694, 6.62689974931662635095789969685, 7.51686479596460400303178433673, 8.429630893089533244710293176268, 9.361120626920682552766967596987

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