Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $-0.441 - 0.897i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.4·3-s + (17.9 − 52.9i)5-s + 86.7i·7-s − 134.·9-s − 258. i·11-s − 465.·13-s + (187. − 551. i)15-s + 2.02e3i·17-s − 1.05e3i·19-s + 904. i·21-s + 3.99e3i·23-s + (−2.47e3 − 1.90e3i)25-s − 3.93e3·27-s + 3.24e3i·29-s − 991.·31-s + ⋯
L(s)  = 1  + 0.668·3-s + (0.321 − 0.946i)5-s + 0.669i·7-s − 0.552·9-s − 0.644i·11-s − 0.763·13-s + (0.215 − 0.633i)15-s + 1.70i·17-s − 0.667i·19-s + 0.447i·21-s + 1.57i·23-s + (−0.792 − 0.609i)25-s − 1.03·27-s + 0.715i·29-s − 0.185·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $-0.441 - 0.897i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ -0.441 - 0.897i)\)
\(L(3)\)  \(\approx\)  \(1.060328986\)
\(L(\frac12)\)  \(\approx\)  \(1.060328986\)
\(L(\frac{7}{2})\)   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 \( 1 + (-17.9 + 52.9i)T \)
good3 \( 1 - 10.4T + 243T^{2} \)
7 \( 1 - 86.7iT - 1.68e4T^{2} \)
11 \( 1 + 258. iT - 1.61e5T^{2} \)
13 \( 1 + 465.T + 3.71e5T^{2} \)
17 \( 1 - 2.02e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.05e3iT - 2.47e6T^{2} \)
23 \( 1 - 3.99e3iT - 6.43e6T^{2} \)
29 \( 1 - 3.24e3iT - 2.05e7T^{2} \)
31 \( 1 + 991.T + 2.86e7T^{2} \)
37 \( 1 + 8.82e3T + 6.93e7T^{2} \)
41 \( 1 - 5.63e3T + 1.15e8T^{2} \)
43 \( 1 - 1.77e3T + 1.47e8T^{2} \)
47 \( 1 - 1.77e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.59e4T + 4.18e8T^{2} \)
59 \( 1 + 1.81e4iT - 7.14e8T^{2} \)
61 \( 1 - 3.54e4iT - 8.44e8T^{2} \)
67 \( 1 - 4.20e4T + 1.35e9T^{2} \)
71 \( 1 + 7.13e4T + 1.80e9T^{2} \)
73 \( 1 + 3.25e3iT - 2.07e9T^{2} \)
79 \( 1 + 8.42e4T + 3.07e9T^{2} \)
83 \( 1 + 7.90e4T + 3.93e9T^{2} \)
89 \( 1 + 6.51e4T + 5.58e9T^{2} \)
97 \( 1 - 3.39e4iT - 8.58e9T^{2} \)
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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

−11.17303949548038027157446438820, −9.940042538050159097263702381869, −8.900501306034852865217346816036, −8.623841518584268144180265921490, −7.52183039864208070592866918411, −5.93484364142283190825576368630, −5.31416854926881078999906341050, −3.86981461302653905169192424879, −2.63586021042387483919498315591, −1.47091482650619724902993193984, 0.23327935948299229808759598813, 2.19538172380811106092459178259, 2.94627807462231791120575563980, 4.23119117083858640639027041970, 5.55561471689188191014375907591, 6.92336345191099015666111553551, 7.43374106549567326518820430800, 8.620034663726760505675910442709, 9.732919368660450515801109015957, 10.28705339272330739499908650335

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