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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 27.5i·3-s + 25i·5-s − 220.·7-s − 517.·9-s − 64.4i·11-s + 703. i·13-s − 689.·15-s − 207.·17-s + 2.79e3i·19-s − 6.07e3i·21-s − 3.25e3·23-s − 625·25-s − 7.55e3i·27-s − 7.89e3i·29-s + 5.05e3·31-s + ⋯
L(s)  = 1  + 1.76i·3-s + 0.447i·5-s − 1.69·7-s − 2.12·9-s − 0.160i·11-s + 1.15i·13-s − 0.790·15-s − 0.174·17-s + 1.77i·19-s − 3.00i·21-s − 1.28·23-s − 0.200·25-s − 1.99i·27-s − 1.74i·29-s + 0.944·31-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  =  \((2,\ 320,\ (\ :5/2),\ 0.707 + 0.707i)\)
\(L(3)\)  \(\approx\)  \(0.2343872337\)
\(L(\frac12)\)  \(\approx\)  \(0.2343872337\)
\(L(\frac{7}{2})\)   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 \( 1 - 27.5iT - 243T^{2} \)
7 \( 1 + 220.T + 1.68e4T^{2} \)
11 \( 1 + 64.4iT - 1.61e5T^{2} \)
13 \( 1 - 703. iT - 3.71e5T^{2} \)
17 \( 1 + 207.T + 1.41e6T^{2} \)
19 \( 1 - 2.79e3iT - 2.47e6T^{2} \)
23 \( 1 + 3.25e3T + 6.43e6T^{2} \)
29 \( 1 + 7.89e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.05e3T + 2.86e7T^{2} \)
37 \( 1 + 4.91e3iT - 6.93e7T^{2} \)
41 \( 1 - 7.03e3T + 1.15e8T^{2} \)
43 \( 1 - 1.51e4iT - 1.47e8T^{2} \)
47 \( 1 + 9.30e3T + 2.29e8T^{2} \)
53 \( 1 + 4.93e3iT - 4.18e8T^{2} \)
59 \( 1 - 3.84e4iT - 7.14e8T^{2} \)
61 \( 1 - 2.77e4iT - 8.44e8T^{2} \)
67 \( 1 + 7.78e3iT - 1.35e9T^{2} \)
71 \( 1 - 6.01e4T + 1.80e9T^{2} \)
73 \( 1 - 3.63e4T + 2.07e9T^{2} \)
79 \( 1 - 3.67e4T + 3.07e9T^{2} \)
83 \( 1 + 2.11e4iT - 3.93e9T^{2} \)
89 \( 1 + 6.38e4T + 5.58e9T^{2} \)
97 \( 1 + 1.43e4T + 8.58e9T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.43698473667099501978402494066, −10.29141105007692516607312341591, −9.850118835054421149341107438648, −9.278336534324477803847796176978, −8.043361554911932918868437861490, −6.38744190785849311180330305912, −5.86248973056348591287472894299, −4.20099929613023415749168867912, −3.73039266279851375141123258068, −2.57265468847618887543440000508, 0.079952774263708683777656388788, 0.830808234856214328157608729086, 2.36466392102560573839320994628, 3.30313558106956034341992633111, 5.26894719767658969037509690431, 6.39167166142974302497084061080, 6.88564742807117747641745367566, 7.940722721641871322995347860522, 8.855490490445653834982863203643, 9.869405649363101954141173123507

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