Properties

Label 2-40e2-1.1-c1-0-33
Degree $2$
Conductor $1600$
Sign $-1$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
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 − 2·7-s + 9-s − 4·11-s − 4·13-s − 4·19-s − 4·21-s + 2·23-s − 4·27-s − 2·29-s − 8·33-s − 4·37-s − 8·39-s + 2·41-s − 6·43-s + 6·47-s − 3·49-s + 4·53-s − 8·57-s − 12·59-s + 10·61-s − 2·63-s + 14·67-s + 4·69-s − 8·71-s + 8·73-s + 8·77-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s − 0.917·19-s − 0.872·21-s + 0.417·23-s − 0.769·27-s − 0.371·29-s − 1.39·33-s − 0.657·37-s − 1.28·39-s + 0.312·41-s − 0.914·43-s + 0.875·47-s − 3/7·49-s + 0.549·53-s − 1.05·57-s − 1.56·59-s + 1.28·61-s − 0.251·63-s + 1.71·67-s + 0.481·69-s − 0.949·71-s + 0.936·73-s + 0.911·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−8.967258895734726958531544509782, −8.262502731210776367824064534493, −7.54305548864136819651663645237, −6.80908802354797739710427841771, −5.69159422021169010829237812675, −4.78734550040710636078302482654, −3.64757269201492355625935308089, −2.79891915671686915828146088412, −2.15062084477359954953472675308, 0, 2.15062084477359954953472675308, 2.79891915671686915828146088412, 3.64757269201492355625935308089, 4.78734550040710636078302482654, 5.69159422021169010829237812675, 6.80908802354797739710427841771, 7.54305548864136819651663645237, 8.262502731210776367824064534493, 8.967258895734726958531544509782

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