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  − 2·3-s + 25·5-s + 62·7-s − 239·9-s − 144·11-s + 654·13-s − 50·15-s − 1.19e3·17-s + 556·19-s − 124·21-s − 2.18e3·23-s + 625·25-s + 964·27-s + 1.57e3·29-s − 9.66e3·31-s + 288·33-s + 1.55e3·35-s + 3.53e3·37-s − 1.30e3·39-s + 7.46e3·41-s − 7.11e3·43-s − 5.97e3·45-s + 2.82e4·47-s − 1.29e4·49-s + 2.38e3·51-s + 1.30e4·53-s − 3.60e3·55-s + ⋯
L(s)  = 1  − 0.128·3-s + 0.447·5-s + 0.478·7-s − 0.983·9-s − 0.358·11-s + 1.07·13-s − 0.0573·15-s − 0.998·17-s + 0.353·19-s − 0.0613·21-s − 0.860·23-s + 1/5·25-s + 0.254·27-s + 0.348·29-s − 1.80·31-s + 0.0460·33-s + 0.213·35-s + 0.424·37-s − 0.137·39-s + 0.693·41-s − 0.586·43-s − 0.439·45-s + 1.86·47-s − 0.771·49-s + 0.128·51-s + 0.637·53-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 \( 1 - p^{2} T \)
good3 \( 1 + 2 T + p^{5} T^{2} \)
7 \( 1 - 62 T + p^{5} T^{2} \)
11 \( 1 + 144 T + p^{5} T^{2} \)
13 \( 1 - 654 T + p^{5} T^{2} \)
17 \( 1 + 70 p T + p^{5} T^{2} \)
19 \( 1 - 556 T + p^{5} T^{2} \)
23 \( 1 + 2182 T + p^{5} T^{2} \)
29 \( 1 - 1578 T + p^{5} T^{2} \)
31 \( 1 + 9660 T + p^{5} T^{2} \)
37 \( 1 - 3534 T + p^{5} T^{2} \)
41 \( 1 - 182 p T + p^{5} T^{2} \)
43 \( 1 + 7114 T + p^{5} T^{2} \)
47 \( 1 - 602 p T + p^{5} T^{2} \)
53 \( 1 - 13046 T + p^{5} T^{2} \)
59 \( 1 + 37092 T + p^{5} T^{2} \)
61 \( 1 + 39570 T + p^{5} T^{2} \)
67 \( 1 + 56734 T + p^{5} T^{2} \)
71 \( 1 + 45588 T + p^{5} T^{2} \)
73 \( 1 - 11842 T + p^{5} T^{2} \)
79 \( 1 + 94216 T + p^{5} T^{2} \)
83 \( 1 + 31482 T + p^{5} T^{2} \)
89 \( 1 + 94054 T + p^{5} T^{2} \)
97 \( 1 - 23714 T + p^{5} T^{2} \)
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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

−10.58915813864721296658113148971, −9.222010450531437665954567924327, −8.564810557374357774471688624999, −7.51087886300826522934790152348, −6.17839955053493035465853379039, −5.52385408461892251771139542118, −4.23626247780489438983867205460, −2.84327141789654988101552242768, −1.59234756708045009683039434322, 0, 1.59234756708045009683039434322, 2.84327141789654988101552242768, 4.23626247780489438983867205460, 5.52385408461892251771139542118, 6.17839955053493035465853379039, 7.51087886300826522934790152348, 8.564810557374357774471688624999, 9.222010450531437665954567924327, 10.58915813864721296658113148971

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