Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $0.258 - 0.965i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3.09i·3-s + 25i·5-s + 118.·7-s + 233.·9-s + 355. i·11-s + 370. i·13-s + 77.4·15-s − 586.·17-s + 972. i·19-s − 368. i·21-s + 1.29e3·23-s − 625·25-s − 1.47e3i·27-s − 5.22e3i·29-s − 4.62e3·31-s + ⋯
L(s)  = 1  − 0.198i·3-s + 0.447i·5-s + 0.916·7-s + 0.960·9-s + 0.885i·11-s + 0.608i·13-s + 0.0889·15-s − 0.491·17-s + 0.618i·19-s − 0.182i·21-s + 0.511·23-s − 0.200·25-s − 0.389i·27-s − 1.15i·29-s − 0.863·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $0.258 - 0.965i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (161, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 0.258 - 0.965i)\)
\(L(3)\)  \(\approx\)  \(2.249478723\)
\(L(\frac12)\)  \(\approx\)  \(2.249478723\)
\(L(\frac{7}{2})\)   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 \( 1 + 3.09iT - 243T^{2} \)
7 \( 1 - 118.T + 1.68e4T^{2} \)
11 \( 1 - 355. iT - 1.61e5T^{2} \)
13 \( 1 - 370. iT - 3.71e5T^{2} \)
17 \( 1 + 586.T + 1.41e6T^{2} \)
19 \( 1 - 972. iT - 2.47e6T^{2} \)
23 \( 1 - 1.29e3T + 6.43e6T^{2} \)
29 \( 1 + 5.22e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.62e3T + 2.86e7T^{2} \)
37 \( 1 - 5.29e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.40e4T + 1.15e8T^{2} \)
43 \( 1 - 4.22e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.85e4T + 2.29e8T^{2} \)
53 \( 1 - 2.09e4iT - 4.18e8T^{2} \)
59 \( 1 - 4.85e4iT - 7.14e8T^{2} \)
61 \( 1 - 3.62e3iT - 8.44e8T^{2} \)
67 \( 1 - 1.68e3iT - 1.35e9T^{2} \)
71 \( 1 + 5.33e3T + 1.80e9T^{2} \)
73 \( 1 - 8.05e4T + 2.07e9T^{2} \)
79 \( 1 + 3.92e4T + 3.07e9T^{2} \)
83 \( 1 - 1.83e4iT - 3.93e9T^{2} \)
89 \( 1 - 4.31e4T + 5.58e9T^{2} \)
97 \( 1 - 1.34e5T + 8.58e9T^{2} \)
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

−11.02572892951542105150028380243, −10.08565264301176375402161516661, −9.220019748623568034480823654404, −7.916423186490520103513166786637, −7.24854664045839039536087250329, −6.27094074145324563077267792165, −4.82193844601161131695529763292, −4.03278676792137627856438102343, −2.29152328936853236888435886443, −1.35244061796468085518440507993, 0.62044490724986986135648336148, 1.81178548793451389068686783146, 3.43271261636448983019776471171, 4.65655509568563853520189634270, 5.40542714922771382725771181467, 6.79705442977139896274623067182, 7.84343266970883794753350373607, 8.716104883369037491282051325554, 9.591716785423224710678875535346, 10.85015636673735857963083847748

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