Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 0

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  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(2.092353006\)
\(L(\frac12)\)  \(\approx\)  \(2.092353006\)
\(L(\frac{7}{2})\)   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

−10.92121899242005282658834065444, −9.767732839943906613110137105946, −8.882255288809734446500491021000, −8.204724157132154784036536704484, −6.61620721836107251824970852720, −6.12222660322088345163522825399, −4.78960808382789177666676797732, −3.44621820856246772996315157557, −2.36102056377420643929697814976, −0.798654792291943964743605701991, 0.798654792291943964743605701991, 2.36102056377420643929697814976, 3.44621820856246772996315157557, 4.78960808382789177666676797732, 6.12222660322088345163522825399, 6.61620721836107251824970852720, 8.204724157132154784036536704484, 8.882255288809734446500491021000, 9.767732839943906613110137105946, 10.92121899242005282658834065444

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