Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $0.948 - 0.317i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.48i·3-s + (53.0 − 17.7i)5-s + 188. i·7-s + 212.·9-s + 501.·11-s − 1.06e3i·13-s + (97.5 + 290. i)15-s + 29.5i·17-s + 1.57e3·19-s − 1.03e3·21-s − 1.29e3i·23-s + (2.49e3 − 1.88e3i)25-s + 2.50e3i·27-s − 3.58e3·29-s − 3.52e3·31-s + ⋯
L(s)  = 1  + 0.352i·3-s + (0.948 − 0.317i)5-s + 1.45i·7-s + 0.875·9-s + 1.25·11-s − 1.74i·13-s + (0.111 + 0.333i)15-s + 0.0248i·17-s + 1.00·19-s − 0.513·21-s − 0.510i·23-s + (0.798 − 0.602i)25-s + 0.660i·27-s − 0.791·29-s − 0.659·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.317i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.948 - 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $0.948 - 0.317i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (129, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 0.948 - 0.317i)\)
\(L(3)\)  \(\approx\)  \(3.120728137\)
\(L(\frac12)\)  \(\approx\)  \(3.120728137\)
\(L(\frac{7}{2})\)   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 \( 1 + (-53.0 + 17.7i)T \)
good3 \( 1 - 5.48iT - 243T^{2} \)
7 \( 1 - 188. iT - 1.68e4T^{2} \)
11 \( 1 - 501.T + 1.61e5T^{2} \)
13 \( 1 + 1.06e3iT - 3.71e5T^{2} \)
17 \( 1 - 29.5iT - 1.41e6T^{2} \)
19 \( 1 - 1.57e3T + 2.47e6T^{2} \)
23 \( 1 + 1.29e3iT - 6.43e6T^{2} \)
29 \( 1 + 3.58e3T + 2.05e7T^{2} \)
31 \( 1 + 3.52e3T + 2.86e7T^{2} \)
37 \( 1 - 8.41e3iT - 6.93e7T^{2} \)
41 \( 1 - 7.01e3T + 1.15e8T^{2} \)
43 \( 1 + 2.26e4iT - 1.47e8T^{2} \)
47 \( 1 - 3.50e3iT - 2.29e8T^{2} \)
53 \( 1 + 2.73e4iT - 4.18e8T^{2} \)
59 \( 1 - 7.92e3T + 7.14e8T^{2} \)
61 \( 1 - 7.02e3T + 8.44e8T^{2} \)
67 \( 1 - 1.76e4iT - 1.35e9T^{2} \)
71 \( 1 - 1.34e4T + 1.80e9T^{2} \)
73 \( 1 - 3.99e4iT - 2.07e9T^{2} \)
79 \( 1 - 9.33e4T + 3.07e9T^{2} \)
83 \( 1 - 5.84e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.39e4T + 5.58e9T^{2} \)
97 \( 1 - 1.10e5iT - 8.58e9T^{2} \)
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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

−10.67747509734756700068493096172, −9.716184812006857286295335080611, −9.199384665345287373964807446812, −8.229979366915336195616627796983, −6.82313492890486375246897952468, −5.68879145519129765151770323853, −5.14211558255885672054951842342, −3.58383262560033827697317904648, −2.27778683286772488784550700867, −1.05419055146902240813976103219, 1.12921163313795282192718778988, 1.80008508909691968244682749483, 3.68043330871409290085833821825, 4.51634894837720179094761513677, 6.12935541128107659583575004588, 6.98990527053543672241596604787, 7.44346197171841642916705740582, 9.358372337969513176797100263937, 9.551656701768483531445567207716, 10.77691648912305896859637642383

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