Properties

Label 2-1288-1.1-c1-0-12
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 7-s + 9-s + 4·11-s + 6·13-s − 2·21-s + 23-s − 5·25-s − 4·27-s + 2·29-s + 2·31-s + 8·33-s + 2·37-s + 12·39-s − 6·41-s + 4·43-s − 2·47-s + 49-s + 14·53-s + 14·59-s + 12·61-s − 63-s − 4·67-s + 2·69-s − 2·73-s − 10·75-s − 4·77-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.436·21-s + 0.208·23-s − 25-s − 0.769·27-s + 0.371·29-s + 0.359·31-s + 1.39·33-s + 0.328·37-s + 1.92·39-s − 0.937·41-s + 0.609·43-s − 0.291·47-s + 1/7·49-s + 1.92·53-s + 1.82·59-s + 1.53·61-s − 0.125·63-s − 0.488·67-s + 0.240·69-s − 0.234·73-s − 1.15·75-s − 0.455·77-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: $\chi_{1288} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1288,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.623016947\)
\(L(\frac12)\) \(\approx\) \(2.623016947\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−9.515159914427731296818870870826, −8.609158405448298113744621998852, −8.469497890062852666766750103040, −7.25741910042798485514816215730, −6.42281710859097363087217391295, −5.62299425528511656843542759056, −4.00727605955274724755807765716, −3.65861145648623169902938937040, −2.51899572530787692578329494636, −1.27179858482475471748824662576, 1.27179858482475471748824662576, 2.51899572530787692578329494636, 3.65861145648623169902938937040, 4.00727605955274724755807765716, 5.62299425528511656843542759056, 6.42281710859097363087217391295, 7.25741910042798485514816215730, 8.469497890062852666766750103040, 8.609158405448298113744621998852, 9.515159914427731296818870870826

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