Properties

Label 2-640-40.19-c0-0-1
Degree $2$
Conductor $640$
Sign $1$
Analytic cond. $0.319401$
Root an. cond. $0.565156$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 9-s + 2·13-s + 25-s − 2·37-s − 2·41-s − 45-s − 49-s + 2·53-s − 2·65-s + 81-s − 2·89-s + 2·117-s + ⋯
L(s)  = 1  − 5-s + 9-s + 2·13-s + 25-s − 2·37-s − 2·41-s − 45-s − 49-s + 2·53-s − 2·65-s + 81-s − 2·89-s + 2·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $1$
Analytic conductor: \(0.319401\)
Root analytic conductor: \(0.565156\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{640} (319, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8943906129\)
\(L(\frac12)\) \(\approx\) \(0.8943906129\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.79495837168484138612076670698, −10.09874468729156596722724484143, −8.785888650409807130869408415447, −8.310120984203962583004789925280, −7.19903958631570605268935541489, −6.51778033020758742396704834596, −5.21270997532034263218530541523, −4.04457968765298506877627190712, −3.41835731076545275239064437262, −1.45578699018930483007679514468, 1.45578699018930483007679514468, 3.41835731076545275239064437262, 4.04457968765298506877627190712, 5.21270997532034263218530541523, 6.51778033020758742396704834596, 7.19903958631570605268935541489, 8.310120984203962583004789925280, 8.785888650409807130869408415447, 10.09874468729156596722724484143, 10.79495837168484138612076670698

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