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  + 28.7·3-s − 25·5-s − 42.1·7-s + 581.·9-s − 416.·11-s − 966.·13-s − 717.·15-s − 1.83e3·17-s − 317.·19-s − 1.21e3·21-s + 1.56e3·23-s + 625·25-s + 9.72e3·27-s − 7.75e3·29-s + 102.·31-s − 1.19e4·33-s + 1.05e3·35-s − 1.93e3·37-s − 2.77e4·39-s + 7.99e3·41-s − 1.65e4·43-s − 1.45e4·45-s + 1.86e4·47-s − 1.50e4·49-s − 5.26e4·51-s + 1.49e4·53-s + 1.04e4·55-s + ⋯
L(s)  = 1  + 1.84·3-s − 0.447·5-s − 0.325·7-s + 2.39·9-s − 1.03·11-s − 1.58·13-s − 0.823·15-s − 1.53·17-s − 0.201·19-s − 0.598·21-s + 0.618·23-s + 0.200·25-s + 2.56·27-s − 1.71·29-s + 0.0191·31-s − 1.91·33-s + 0.145·35-s − 0.232·37-s − 2.92·39-s + 0.742·41-s − 1.36·43-s − 1.07·45-s + 1.23·47-s − 0.894·49-s − 2.83·51-s + 0.732·53-s + 0.463·55-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 \( 1 - 28.7T + 243T^{2} \)
7 \( 1 + 42.1T + 1.68e4T^{2} \)
11 \( 1 + 416.T + 1.61e5T^{2} \)
13 \( 1 + 966.T + 3.71e5T^{2} \)
17 \( 1 + 1.83e3T + 1.41e6T^{2} \)
19 \( 1 + 317.T + 2.47e6T^{2} \)
23 \( 1 - 1.56e3T + 6.43e6T^{2} \)
29 \( 1 + 7.75e3T + 2.05e7T^{2} \)
31 \( 1 - 102.T + 2.86e7T^{2} \)
37 \( 1 + 1.93e3T + 6.93e7T^{2} \)
41 \( 1 - 7.99e3T + 1.15e8T^{2} \)
43 \( 1 + 1.65e4T + 1.47e8T^{2} \)
47 \( 1 - 1.86e4T + 2.29e8T^{2} \)
53 \( 1 - 1.49e4T + 4.18e8T^{2} \)
59 \( 1 + 1.98e4T + 7.14e8T^{2} \)
61 \( 1 - 1.80e4T + 8.44e8T^{2} \)
67 \( 1 - 5.50e4T + 1.35e9T^{2} \)
71 \( 1 - 1.12e4T + 1.80e9T^{2} \)
73 \( 1 + 4.01e3T + 2.07e9T^{2} \)
79 \( 1 - 2.40e4T + 3.07e9T^{2} \)
83 \( 1 + 7.05e4T + 3.93e9T^{2} \)
89 \( 1 + 6.07e4T + 5.58e9T^{2} \)
97 \( 1 + 3.11e4T + 8.58e9T^{2} \)
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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

−10.01922427600089407112836379233, −9.282235898816320923547648625365, −8.456292417091123377035563213955, −7.55210304967867538391863570003, −6.96139724634121783584378548629, −4.98564328454176858249239791859, −3.94496176282658847171563747832, −2.78732980976682471581089184963, −2.10645138909285204255257630129, 0, 2.10645138909285204255257630129, 2.78732980976682471581089184963, 3.94496176282658847171563747832, 4.98564328454176858249239791859, 6.96139724634121783584378548629, 7.55210304967867538391863570003, 8.456292417091123377035563213955, 9.282235898816320923547648625365, 10.01922427600089407112836379233

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