Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $-0.707 - 0.707i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 21.6i·3-s − 25i·5-s + 169.·7-s − 224.·9-s + 512. i·11-s − 36.4i·13-s − 540.·15-s − 1.26e3·17-s − 790. i·19-s − 3.66e3i·21-s − 4.99e3·23-s − 625·25-s − 401. i·27-s + 934. i·29-s − 6.69e3·31-s + ⋯
L(s)  = 1  − 1.38i·3-s − 0.447i·5-s + 1.30·7-s − 0.923·9-s + 1.27i·11-s − 0.0598i·13-s − 0.620·15-s − 1.05·17-s − 0.502i·19-s − 1.81i·21-s − 1.96·23-s − 0.200·25-s − 0.106i·27-s + 0.206i·29-s − 1.25·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $-0.707 - 0.707i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (161, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ -0.707 - 0.707i)\)
\(L(3)\)  \(\approx\)  \(0.6156366010\)
\(L(\frac12)\)  \(\approx\)  \(0.6156366010\)
\(L(\frac{7}{2})\)   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 \( 1 + 25iT \)
good3 \( 1 + 21.6iT - 243T^{2} \)
7 \( 1 - 169.T + 1.68e4T^{2} \)
11 \( 1 - 512. iT - 1.61e5T^{2} \)
13 \( 1 + 36.4iT - 3.71e5T^{2} \)
17 \( 1 + 1.26e3T + 1.41e6T^{2} \)
19 \( 1 + 790. iT - 2.47e6T^{2} \)
23 \( 1 + 4.99e3T + 6.43e6T^{2} \)
29 \( 1 - 934. iT - 2.05e7T^{2} \)
31 \( 1 + 6.69e3T + 2.86e7T^{2} \)
37 \( 1 + 4.30e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.03e4T + 1.15e8T^{2} \)
43 \( 1 + 6.37e3iT - 1.47e8T^{2} \)
47 \( 1 + 2.21e4T + 2.29e8T^{2} \)
53 \( 1 + 2.32e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.77e4iT - 7.14e8T^{2} \)
61 \( 1 - 3.99e4iT - 8.44e8T^{2} \)
67 \( 1 + 5.54e4iT - 1.35e9T^{2} \)
71 \( 1 + 3.98e4T + 1.80e9T^{2} \)
73 \( 1 + 435.T + 2.07e9T^{2} \)
79 \( 1 - 6.16e4T + 3.07e9T^{2} \)
83 \( 1 + 2.94e3iT - 3.93e9T^{2} \)
89 \( 1 - 6.47e4T + 5.58e9T^{2} \)
97 \( 1 - 1.59e5T + 8.58e9T^{2} \)
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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

−10.29974353143131261636943514334, −9.013121586745250510204705669385, −8.066691615613525683869735538418, −7.43309161014060523779783333450, −6.51009672101509303532685526640, −5.20905668715709231077410322895, −4.23760458531656687663261283452, −2.04381161048398273226901902166, −1.70949622896718845685421297548, −0.14598088814944597392378694741, 1.85821344456188905099223676832, 3.41595835980484164835800013252, 4.29531343749330299949338811906, 5.26960447849903358472282259007, 6.30352600845563927318296074689, 7.895374565857518298028218952169, 8.584312376961287102788224686056, 9.637494364741441410158247477031, 10.58380712704199638978183070831, 11.14758314622526670044298465550

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