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  − 16.7·3-s − 25·5-s + 94.1·7-s + 36.4·9-s − 143.·11-s − 421.·13-s + 417.·15-s + 1.98e3·17-s + 1.31e3·19-s − 1.57e3·21-s − 4.02e3·23-s + 625·25-s + 3.45e3·27-s + 6.41e3·29-s − 2.35e3·31-s + 2.40e3·33-s − 2.35e3·35-s + 7.87e3·37-s + 7.04e3·39-s + 1.50e4·41-s − 1.14e3·43-s − 910.·45-s − 2.15e4·47-s − 7.94e3·49-s − 3.31e4·51-s − 9.56e3·53-s + 3.59e3·55-s + ⋯
L(s)  = 1  − 1.07·3-s − 0.447·5-s + 0.726·7-s + 0.149·9-s − 0.358·11-s − 0.691·13-s + 0.479·15-s + 1.66·17-s + 0.837·19-s − 0.778·21-s − 1.58·23-s + 0.200·25-s + 0.911·27-s + 1.41·29-s − 0.439·31-s + 0.383·33-s − 0.324·35-s + 0.945·37-s + 0.741·39-s + 1.40·41-s − 0.0941·43-s − 0.0670·45-s − 1.42·47-s − 0.472·49-s − 1.78·51-s − 0.467·53-s + 0.160·55-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 \( 1 + 16.7T + 243T^{2} \)
7 \( 1 - 94.1T + 1.68e4T^{2} \)
11 \( 1 + 143.T + 1.61e5T^{2} \)
13 \( 1 + 421.T + 3.71e5T^{2} \)
17 \( 1 - 1.98e3T + 1.41e6T^{2} \)
19 \( 1 - 1.31e3T + 2.47e6T^{2} \)
23 \( 1 + 4.02e3T + 6.43e6T^{2} \)
29 \( 1 - 6.41e3T + 2.05e7T^{2} \)
31 \( 1 + 2.35e3T + 2.86e7T^{2} \)
37 \( 1 - 7.87e3T + 6.93e7T^{2} \)
41 \( 1 - 1.50e4T + 1.15e8T^{2} \)
43 \( 1 + 1.14e3T + 1.47e8T^{2} \)
47 \( 1 + 2.15e4T + 2.29e8T^{2} \)
53 \( 1 + 9.56e3T + 4.18e8T^{2} \)
59 \( 1 + 4.27e4T + 7.14e8T^{2} \)
61 \( 1 + 3.21e4T + 8.44e8T^{2} \)
67 \( 1 - 3.03e4T + 1.35e9T^{2} \)
71 \( 1 - 3.60e4T + 1.80e9T^{2} \)
73 \( 1 + 6.34e4T + 2.07e9T^{2} \)
79 \( 1 + 8.99e4T + 3.07e9T^{2} \)
83 \( 1 + 3.82e4T + 3.93e9T^{2} \)
89 \( 1 - 5.74e3T + 5.58e9T^{2} \)
97 \( 1 - 1.78e5T + 8.58e9T^{2} \)
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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

−10.45947058434478772022474981015, −9.657531629097332602211321910078, −8.103754572222881226220997678658, −7.60267796217372383669773747510, −6.20229177997926447256767719308, −5.33989874029378919615111521975, −4.49594351488638667210590265303, −2.96809213773202582632171074880, −1.23787085294034386920698253609, 0, 1.23787085294034386920698253609, 2.96809213773202582632171074880, 4.49594351488638667210590265303, 5.33989874029378919615111521975, 6.20229177997926447256767719308, 7.60267796217372383669773747510, 8.103754572222881226220997678658, 9.657531629097332602211321910078, 10.45947058434478772022474981015

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