Properties

Degree 2
Conductor $ 2^{4} \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

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.317i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.948 - 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(80\)    =    \(2^{4} \cdot 5\)
\( \varepsilon \)  =  $0.948 - 0.317i$
motivic weight  =  \(5\)
character  :  $\chi_{80} (49, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 80,\ (\ :5/2),\ 0.948 - 0.317i)\)
\(L(3)\)  \(\approx\)  \(1.69225 + 0.275984i\)
\(L(\frac12)\)  \(\approx\)  \(1.69225 + 0.275984i\)
\(L(\frac{7}{2})\)   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

−13.77101024672805608138069052579, −12.13831260558845063625261514145, −11.30171205979825034153443006169, −10.19322846643656677612947770281, −9.085761347630710113054138397819, −7.36127340579589979226496778453, −6.82779675926616974137684144517, −4.32848171231335642919762942709, −3.84390793311885044065424965376, −1.13852016970987412421399429156, 1.03325997884725435930193708824, 3.09247284878212337570664882686, 4.78301319564271191630490035879, 6.25863788812066681634698420121, 7.71214874150787769759069194937, 8.656974663170402625617457534732, 9.906158210197541117424359718909, 11.54715113101890016376144835783, 12.25858135021787745605874680191, 13.00596437079220066239822225447

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