Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 20.7·3-s − 25·5-s + 35.2·7-s + 186.·9-s + 7.33·11-s − 619.·13-s + 518.·15-s + 959.·17-s + 309.·19-s − 731.·21-s + 2.46e3·23-s + 625·25-s + 1.16e3·27-s + 1.28e3·29-s + 7.09e3·31-s − 152.·33-s − 881.·35-s + 6.10e3·37-s + 1.28e4·39-s − 1.88e4·41-s − 3.14e3·43-s − 4.67e3·45-s + 2.05e4·47-s − 1.55e4·49-s − 1.98e4·51-s − 3.37e4·53-s − 183.·55-s + ⋯
L(s)  = 1  − 1.33·3-s − 0.447·5-s + 0.272·7-s + 0.768·9-s + 0.0182·11-s − 1.01·13-s + 0.594·15-s + 0.805·17-s + 0.196·19-s − 0.361·21-s + 0.972·23-s + 0.200·25-s + 0.307·27-s + 0.283·29-s + 1.32·31-s − 0.0242·33-s − 0.121·35-s + 0.732·37-s + 1.35·39-s − 1.74·41-s − 0.259·43-s − 0.343·45-s + 1.35·47-s − 0.925·49-s − 1.07·51-s − 1.64·53-s − 0.00817·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(5\)
character  :  $\chi_{320} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ -1)\)
\(L(3)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 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 \( 1 + 25T \)
good3 \( 1 + 20.7T + 243T^{2} \)
7 \( 1 - 35.2T + 1.68e4T^{2} \)
11 \( 1 - 7.33T + 1.61e5T^{2} \)
13 \( 1 + 619.T + 3.71e5T^{2} \)
17 \( 1 - 959.T + 1.41e6T^{2} \)
19 \( 1 - 309.T + 2.47e6T^{2} \)
23 \( 1 - 2.46e3T + 6.43e6T^{2} \)
29 \( 1 - 1.28e3T + 2.05e7T^{2} \)
31 \( 1 - 7.09e3T + 2.86e7T^{2} \)
37 \( 1 - 6.10e3T + 6.93e7T^{2} \)
41 \( 1 + 1.88e4T + 1.15e8T^{2} \)
43 \( 1 + 3.14e3T + 1.47e8T^{2} \)
47 \( 1 - 2.05e4T + 2.29e8T^{2} \)
53 \( 1 + 3.37e4T + 4.18e8T^{2} \)
59 \( 1 + 1.50e4T + 7.14e8T^{2} \)
61 \( 1 - 7.54e3T + 8.44e8T^{2} \)
67 \( 1 + 2.55e4T + 1.35e9T^{2} \)
71 \( 1 + 5.62e4T + 1.80e9T^{2} \)
73 \( 1 - 5.86e4T + 2.07e9T^{2} \)
79 \( 1 - 3.22e4T + 3.07e9T^{2} \)
83 \( 1 + 3.11e4T + 3.93e9T^{2} \)
89 \( 1 + 7.52e4T + 5.58e9T^{2} \)
97 \( 1 + 1.76e5T + 8.58e9T^{2} \)
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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

−10.50496709389901207912464978883, −9.652667233061459678777748006780, −8.293515125729667654715865091733, −7.29175737143444691659665823011, −6.35975843724824613032659578136, −5.23557289118580835816384771206, −4.58590026043991890051644622714, −2.98303800976353749216956902101, −1.15654369443519261311370299540, 0, 1.15654369443519261311370299540, 2.98303800976353749216956902101, 4.58590026043991890051644622714, 5.23557289118580835816384771206, 6.35975843724824613032659578136, 7.29175737143444691659665823011, 8.293515125729667654715865091733, 9.652667233061459678777748006780, 10.50496709389901207912464978883

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