Properties

Label 2-3200-1.1-c1-0-72
Degree $2$
Conductor $3200$
Sign $1$
Analytic cond. $25.5521$
Root an. cond. $5.05491$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 4·7-s + 6·9-s − 3·11-s − 2·13-s − 3·17-s − 7·19-s + 12·21-s − 6·23-s − 9·27-s − 6·29-s + 10·31-s + 9·33-s + 6·39-s − 11·41-s − 4·43-s + 2·47-s + 9·49-s + 9·51-s − 14·53-s + 21·57-s − 4·59-s − 8·61-s − 24·63-s − 5·67-s + 18·69-s − 2·71-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.51·7-s + 2·9-s − 0.904·11-s − 0.554·13-s − 0.727·17-s − 1.60·19-s + 2.61·21-s − 1.25·23-s − 1.73·27-s − 1.11·29-s + 1.79·31-s + 1.56·33-s + 0.960·39-s − 1.71·41-s − 0.609·43-s + 0.291·47-s + 9/7·49-s + 1.26·51-s − 1.92·53-s + 2.78·57-s − 0.520·59-s − 1.02·61-s − 3.02·63-s − 0.610·67-s + 2.16·69-s − 0.237·71-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
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 \( 1 + p T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−7.71326464716231478539436675643, −6.82434328183666949842646575912, −6.28202429547059167627744457953, −5.89574155606320244691885649601, −4.84234041807494014000224350092, −4.28953538947155706850960860349, −3.08851197027170221294368941550, −1.91885855887521311719438128799, 0, 0, 1.91885855887521311719438128799, 3.08851197027170221294368941550, 4.28953538947155706850960860349, 4.84234041807494014000224350092, 5.89574155606320244691885649601, 6.28202429547059167627744457953, 6.82434328183666949842646575912, 7.71326464716231478539436675643

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