Properties

Label 2-3200-1.1-c1-0-58
Degree $2$
Conductor $3200$
Sign $-1$
Analytic cond. $25.5521$
Root an. cond. $5.05491$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.806·3-s + 2.15·7-s − 2.35·9-s + 0.387·11-s + 0.962·13-s + 1.61·17-s − 6.31·19-s − 1.73·21-s − 6.15·23-s + 4.31·27-s + 2·29-s + 9.92·31-s − 0.312·33-s − 6.57·37-s − 0.775·39-s + 4.57·41-s − 11.5·43-s − 4.54·47-s − 2.35·49-s − 1.29·51-s − 8.96·53-s + 5.08·57-s + 6.31·59-s + 0.261·61-s − 5.06·63-s − 9.89·67-s + 4.96·69-s + ⋯
L(s)  = 1  − 0.465·3-s + 0.815·7-s − 0.783·9-s + 0.116·11-s + 0.266·13-s + 0.390·17-s − 1.44·19-s − 0.379·21-s − 1.28·23-s + 0.829·27-s + 0.371·29-s + 1.78·31-s − 0.0544·33-s − 1.08·37-s − 0.124·39-s + 0.714·41-s − 1.75·43-s − 0.662·47-s − 0.335·49-s − 0.181·51-s − 1.23·53-s + 0.673·57-s + 0.821·59-s + 0.0335·61-s − 0.638·63-s − 1.20·67-s + 0.597·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(25.5521\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3200,\ (\ :1/2),\ -1)\)

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 \( 1 + 0.806T + 3T^{2} \)
7 \( 1 - 2.15T + 7T^{2} \)
11 \( 1 - 0.387T + 11T^{2} \)
13 \( 1 - 0.962T + 13T^{2} \)
17 \( 1 - 1.61T + 17T^{2} \)
19 \( 1 + 6.31T + 19T^{2} \)
23 \( 1 + 6.15T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 9.92T + 31T^{2} \)
37 \( 1 + 6.57T + 37T^{2} \)
41 \( 1 - 4.57T + 41T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 + 4.54T + 47T^{2} \)
53 \( 1 + 8.96T + 53T^{2} \)
59 \( 1 - 6.31T + 59T^{2} \)
61 \( 1 - 0.261T + 61T^{2} \)
67 \( 1 + 9.89T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 - 13.0T + 73T^{2} \)
79 \( 1 + 1.92T + 79T^{2} \)
83 \( 1 + 2.88T + 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 - 9.61T + 97T^{2} \)
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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

−8.341646904588237869586721934434, −7.74329040001408118332557337193, −6.43488057214508888743285331245, −6.23508442497104636865024695042, −5.14328100056141709087713094219, −4.58414961466488286491578627007, −3.57711524724079370275944885397, −2.48389997533222473437659389103, −1.45879773908393767285652525300, 0, 1.45879773908393767285652525300, 2.48389997533222473437659389103, 3.57711524724079370275944885397, 4.58414961466488286491578627007, 5.14328100056141709087713094219, 6.23508442497104636865024695042, 6.43488057214508888743285331245, 7.74329040001408118332557337193, 8.341646904588237869586721934434

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