Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $1$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 30.1i·3-s + 55.9·5-s + 194. i·7-s − 667.·9-s − 1.68e3i·15-s + 5.86e3·21-s + 5.06e3i·23-s + 3.12e3·25-s + 1.28e4i·27-s − 1.68e3·29-s + 1.08e4i·35-s + 2.10e4·41-s + 1.01e4i·43-s − 3.73e4·45-s + 3.21e3i·47-s + ⋯
L(s)  = 1  − 1.93i·3-s + 0.999·5-s + 1.49i·7-s − 2.74·9-s − 1.93i·15-s + 2.89·21-s + 1.99i·23-s + 25-s + 3.38i·27-s − 0.372·29-s + 1.49i·35-s + 1.95·41-s + 0.837i·43-s − 2.74·45-s + 0.212i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{320} (129, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(1.972645972\)
\(L(\frac12)\)  \(\approx\)  \(1.972645972\)
\(L(\frac{7}{2})\)   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 \( 1 - 55.9T \)
good3 \( 1 + 30.1iT - 243T^{2} \)
7 \( 1 - 194. iT - 1.68e4T^{2} \)
11 \( 1 + 1.61e5T^{2} \)
13 \( 1 - 3.71e5T^{2} \)
17 \( 1 - 1.41e6T^{2} \)
19 \( 1 + 2.47e6T^{2} \)
23 \( 1 - 5.06e3iT - 6.43e6T^{2} \)
29 \( 1 + 1.68e3T + 2.05e7T^{2} \)
31 \( 1 + 2.86e7T^{2} \)
37 \( 1 - 6.93e7T^{2} \)
41 \( 1 - 2.10e4T + 1.15e8T^{2} \)
43 \( 1 - 1.01e4iT - 1.47e8T^{2} \)
47 \( 1 - 3.21e3iT - 2.29e8T^{2} \)
53 \( 1 - 4.18e8T^{2} \)
59 \( 1 + 7.14e8T^{2} \)
61 \( 1 - 5.22e4T + 8.44e8T^{2} \)
67 \( 1 - 6.35e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.80e9T^{2} \)
73 \( 1 - 2.07e9T^{2} \)
79 \( 1 + 3.07e9T^{2} \)
83 \( 1 + 1.16e5iT - 3.93e9T^{2} \)
89 \( 1 + 1.49e5T + 5.58e9T^{2} \)
97 \( 1 - 8.58e9T^{2} \)
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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

−11.23840591002921423643807703180, −9.535184176654856440977763745549, −8.826857318040689824736983022122, −7.85771784620067031442391585798, −6.87776193681051029686958129355, −5.80513218378867535521654327306, −5.56089941471565918349886250617, −2.88549434951700315025926441153, −2.10207849389218407270804125482, −1.20492241706955961691313177704, 0.53770298992058534174940357216, 2.59665113258588841773122841067, 3.87911010103700635423855442789, 4.58326230182219684256444453552, 5.60110135330737652909960761924, 6.72737497975470380956938186711, 8.279552753779466766791288894968, 9.271522827850503363751293984789, 9.994548441109771740520175238692, 10.61767889651437474590897271727

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