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Properties

Label 2-1280-20.19-c0-0-0
Degree $2$
Conductor $1280$
Sign $1$
Analytic cond. $0.638803$
Root an. cond. $0.799251$
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  − 1.41·3-s − 5-s − 1.41·7-s + 1.00·9-s + 1.41·15-s + 2.00·21-s + 1.41·23-s + 25-s + 1.41·35-s + 1.41·43-s − 1.00·45-s − 1.41·47-s + 1.00·49-s + 2·61-s − 1.41·63-s + 1.41·67-s − 2.00·69-s − 1.41·75-s − 0.999·81-s + 1.41·83-s − 2·89-s + 1.41·103-s − 2.00·105-s − 1.41·107-s − 2·109-s − 1.41·115-s + ⋯
L(s)  = 1  − 1.41·3-s − 5-s − 1.41·7-s + 1.00·9-s + 1.41·15-s + 2.00·21-s + 1.41·23-s + 25-s + 1.41·35-s + 1.41·43-s − 1.00·45-s − 1.41·47-s + 1.00·49-s + 2·61-s − 1.41·63-s + 1.41·67-s − 2.00·69-s − 1.41·75-s − 0.999·81-s + 1.41·83-s − 2·89-s + 1.41·103-s − 2.00·105-s − 1.41·107-s − 2·109-s − 1.41·115-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $1$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (1279, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3948154774\)
\(L(\frac12)\) \(\approx\) \(0.3948154774\)
\(L(1)\) 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 \( 1 + 1.41T + T^{2} \)
7 \( 1 + 1.41T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.41T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 1.41T + T^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 2T + T^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 + 2T + T^{2} \)
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

−9.989399383115153507899038229426, −9.202713195724998338614066340546, −8.184355127075122758496533876636, −6.99704207618037254120184822889, −6.70421426577398929925581216387, −5.71458161335562822697603802377, −4.88521014165427488349416963154, −3.86772964661671157802939162997, −2.92100914359994698244124040325, −0.71517719326576147399506352774, 0.71517719326576147399506352774, 2.92100914359994698244124040325, 3.86772964661671157802939162997, 4.88521014165427488349416963154, 5.71458161335562822697603802377, 6.70421426577398929925581216387, 6.99704207618037254120184822889, 8.184355127075122758496533876636, 9.202713195724998338614066340546, 9.989399383115153507899038229426

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