Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 5-s − 2·7-s + 9-s − 2·13-s − 2·15-s − 6·17-s − 4·19-s + 4·21-s − 6·23-s + 25-s + 4·27-s − 6·29-s + 4·31-s − 2·35-s − 2·37-s + 4·39-s + 6·41-s − 10·43-s + 45-s + 6·47-s − 3·49-s + 12·51-s + 6·53-s + 8·57-s + 12·59-s − 2·61-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.554·13-s − 0.516·15-s − 1.45·17-s − 0.917·19-s + 0.872·21-s − 1.25·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.338·35-s − 0.328·37-s + 0.640·39-s + 0.937·41-s − 1.52·43-s + 0.149·45-s + 0.875·47-s − 3/7·49-s + 1.68·51-s + 0.824·53-s + 1.05·57-s + 1.56·59-s − 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{320} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 320,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.16454648418111686213251777526, −10.35239107507603235483123434521, −9.511328653906225471558818278712, −8.404829898294941223649273971532, −6.87772159023272676690262517558, −6.26903409609752801893873868553, −5.32703134203753576211066117067, −4.13045861856120535222472022668, −2.32736297473362010207341859714, 0, 2.32736297473362010207341859714, 4.13045861856120535222472022668, 5.32703134203753576211066117067, 6.26903409609752801893873868553, 6.87772159023272676690262517558, 8.404829898294941223649273971532, 9.511328653906225471558818278712, 10.35239107507603235483123434521, 11.16454648418111686213251777526

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