Properties

Label 2-2576-1.1-c1-0-34
Degree $2$
Conductor $2576$
Sign $-1$
Analytic cond. $20.5694$
Root an. cond. $4.53535$
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 − 4·11-s + 6·17-s + 6·19-s + 2·21-s + 23-s − 5·25-s + 4·27-s + 10·29-s − 4·31-s + 8·33-s − 2·37-s − 10·41-s + 4·43-s − 12·47-s + 49-s − 12·51-s − 6·53-s − 12·57-s + 2·59-s − 63-s − 2·69-s + 8·71-s − 6·73-s + 10·75-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.45·17-s + 1.37·19-s + 0.436·21-s + 0.208·23-s − 25-s + 0.769·27-s + 1.85·29-s − 0.718·31-s + 1.39·33-s − 0.328·37-s − 1.56·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s − 1.68·51-s − 0.824·53-s − 1.58·57-s + 0.260·59-s − 0.125·63-s − 0.240·69-s + 0.949·71-s − 0.702·73-s + 1.15·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(20.5694\)
Root analytic conductor: \(4.53535\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2576} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2576,\ (\ :1/2),\ -1)\)

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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
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

−8.301744712687195431760292309539, −7.74988710039643353360575497141, −6.85177319222593067310879867892, −6.08177121463097384563662103024, −5.24601487899152555441015567209, −5.03388474470559153479602939879, −3.54598166631464413056339917558, −2.79551625971028242736398494921, −1.23146513560978814112389724417, 0, 1.23146513560978814112389724417, 2.79551625971028242736398494921, 3.54598166631464413056339917558, 5.03388474470559153479602939879, 5.24601487899152555441015567209, 6.08177121463097384563662103024, 6.85177319222593067310879867892, 7.74988710039643353360575497141, 8.301744712687195431760292309539

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