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Properties

Label 2-2564-2564.2563-c0-0-8
Degree $2$
Conductor $2564$
Sign $1$
Analytic cond. $1.27960$
Root an. cond. $1.13119$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 0.445·3-s + 4-s + 1.24·5-s − 0.445·6-s + 8-s − 0.801·9-s + 1.24·10-s − 0.445·12-s − 0.445·13-s − 0.554·15-s + 16-s − 0.801·18-s + 2·19-s + 1.24·20-s − 1.80·23-s − 0.445·24-s + 0.554·25-s − 0.445·26-s + 0.801·27-s − 0.554·30-s + 1.24·31-s + 32-s − 0.801·36-s − 1.80·37-s + 2·38-s + 0.198·39-s + ⋯
L(s)  = 1  + 2-s − 0.445·3-s + 4-s + 1.24·5-s − 0.445·6-s + 8-s − 0.801·9-s + 1.24·10-s − 0.445·12-s − 0.445·13-s − 0.554·15-s + 16-s − 0.801·18-s + 2·19-s + 1.24·20-s − 1.80·23-s − 0.445·24-s + 0.554·25-s − 0.445·26-s + 0.801·27-s − 0.554·30-s + 1.24·31-s + 32-s − 0.801·36-s − 1.80·37-s + 2·38-s + 0.198·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2564\)    =    \(2^{2} \cdot 641\)
Sign: $1$
Analytic conductor: \(1.27960\)
Root analytic conductor: \(1.13119\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2564} (2563, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2564,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.370670760\)
\(L(\frac12)\) \(\approx\) \(2.370670760\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
641 \( 1 - T \)
good3 \( 1 + 0.445T + T^{2} \)
5 \( 1 - 1.24T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 0.445T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 2T + T^{2} \)
23 \( 1 + 1.80T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.24T + T^{2} \)
37 \( 1 + 1.80T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.24T + T^{2} \)
47 \( 1 + 1.80T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.80T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.80T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.24T + T^{2} \)
89 \( 1 + 1.80T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.242996571219555254678798961906, −8.158283600884417146167234631820, −7.32632801813618443496203061698, −6.43767414339166619181578951471, −5.75129440885832287795099026504, −5.41741693932143603587122218683, −4.53159405639407484852075102208, −3.30547642187812360278066655044, −2.53364820805870519000159641465, −1.51276049914055872997619117985, 1.51276049914055872997619117985, 2.53364820805870519000159641465, 3.30547642187812360278066655044, 4.53159405639407484852075102208, 5.41741693932143603587122218683, 5.75129440885832287795099026504, 6.43767414339166619181578951471, 7.32632801813618443496203061698, 8.158283600884417146167234631820, 9.242996571219555254678798961906

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