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  − 18·3-s + 25·5-s − 242·7-s + 81·9-s + 656·11-s + 206·13-s − 450·15-s + 1.69e3·17-s − 1.36e3·19-s + 4.35e3·21-s − 2.19e3·23-s + 625·25-s + 2.91e3·27-s + 2.21e3·29-s + 1.70e3·31-s − 1.18e4·33-s − 6.05e3·35-s + 846·37-s − 3.70e3·39-s − 1.81e3·41-s + 1.05e4·43-s + 2.02e3·45-s − 1.20e4·47-s + 4.17e4·49-s − 3.04e4·51-s − 3.25e4·53-s + 1.64e4·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s − 1.86·7-s + 1/3·9-s + 1.63·11-s + 0.338·13-s − 0.516·15-s + 1.41·17-s − 0.866·19-s + 2.15·21-s − 0.866·23-s + 1/5·25-s + 0.769·27-s + 0.489·29-s + 0.317·31-s − 1.88·33-s − 0.834·35-s + 0.101·37-s − 0.390·39-s − 0.168·41-s + 0.868·43-s + 0.149·45-s − 0.797·47-s + 2.48·49-s − 1.63·51-s − 1.59·53-s + 0.731·55-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 \( 1 + 2 p^{2} T + p^{5} T^{2} \)
7 \( 1 + 242 T + p^{5} T^{2} \)
11 \( 1 - 656 T + p^{5} T^{2} \)
13 \( 1 - 206 T + p^{5} T^{2} \)
17 \( 1 - 1690 T + p^{5} T^{2} \)
19 \( 1 + 1364 T + p^{5} T^{2} \)
23 \( 1 + 2198 T + p^{5} T^{2} \)
29 \( 1 - 2218 T + p^{5} T^{2} \)
31 \( 1 - 1700 T + p^{5} T^{2} \)
37 \( 1 - 846 T + p^{5} T^{2} \)
41 \( 1 + 1818 T + p^{5} T^{2} \)
43 \( 1 - 10534 T + p^{5} T^{2} \)
47 \( 1 + 12074 T + p^{5} T^{2} \)
53 \( 1 + 32586 T + p^{5} T^{2} \)
59 \( 1 - 8668 T + p^{5} T^{2} \)
61 \( 1 - 34670 T + p^{5} T^{2} \)
67 \( 1 + 47566 T + p^{5} T^{2} \)
71 \( 1 + 948 T + p^{5} T^{2} \)
73 \( 1 + 63102 T + p^{5} T^{2} \)
79 \( 1 + 46536 T + p^{5} T^{2} \)
83 \( 1 + 88778 T + p^{5} T^{2} \)
89 \( 1 + 104934 T + p^{5} T^{2} \)
97 \( 1 + 36254 T + p^{5} T^{2} \)
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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

−10.17725374368552567446005884750, −9.703086108062566005746241758059, −8.647411502744937884381130655920, −6.94150462852851460269880493442, −6.20944681805223179633840619252, −5.81810057395393420274983185550, −4.18122877122336120612440865362, −3.08078221577963605259660239621, −1.20388656091316167544648215703, 0, 1.20388656091316167544648215703, 3.08078221577963605259660239621, 4.18122877122336120612440865362, 5.81810057395393420274983185550, 6.20944681805223179633840619252, 6.94150462852851460269880493442, 8.647411502744937884381130655920, 9.703086108062566005746241758059, 10.17725374368552567446005884750

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