Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $0.419 - 0.907i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.69i·3-s + (−23.4 + 50.7i)5-s + 10.2i·7-s + 220.·9-s + 596.·11-s + 420. i·13-s + (−238. − 109. i)15-s − 974. i·17-s + 380.·19-s − 48.1·21-s − 3.54e3i·23-s + (−2.02e3 − 2.37e3i)25-s + 2.17e3i·27-s + 5.44e3·29-s + 3.62e3·31-s + ⋯
L(s)  = 1  + 0.301i·3-s + (−0.419 + 0.907i)5-s + 0.0791i·7-s + 0.909·9-s + 1.48·11-s + 0.690i·13-s + (−0.273 − 0.126i)15-s − 0.817i·17-s + 0.241·19-s − 0.0238·21-s − 1.39i·23-s + (−0.648 − 0.760i)25-s + 0.574i·27-s + 1.20·29-s + 0.677·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $0.419 - 0.907i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (129, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 0.419 - 0.907i)\)
\(L(3)\)  \(\approx\)  \(2.350688691\)
\(L(\frac12)\)  \(\approx\)  \(2.350688691\)
\(L(\frac{7}{2})\)   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 \( 1 + (23.4 - 50.7i)T \)
good3 \( 1 - 4.69iT - 243T^{2} \)
7 \( 1 - 10.2iT - 1.68e4T^{2} \)
11 \( 1 - 596.T + 1.61e5T^{2} \)
13 \( 1 - 420. iT - 3.71e5T^{2} \)
17 \( 1 + 974. iT - 1.41e6T^{2} \)
19 \( 1 - 380.T + 2.47e6T^{2} \)
23 \( 1 + 3.54e3iT - 6.43e6T^{2} \)
29 \( 1 - 5.44e3T + 2.05e7T^{2} \)
31 \( 1 - 3.62e3T + 2.86e7T^{2} \)
37 \( 1 - 1.75e3iT - 6.93e7T^{2} \)
41 \( 1 - 263.T + 1.15e8T^{2} \)
43 \( 1 - 1.44e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.34e4iT - 2.29e8T^{2} \)
53 \( 1 + 3.34e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.90e3T + 7.14e8T^{2} \)
61 \( 1 + 2.94e4T + 8.44e8T^{2} \)
67 \( 1 + 7.16e3iT - 1.35e9T^{2} \)
71 \( 1 - 8.13e4T + 1.80e9T^{2} \)
73 \( 1 - 5.51e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.64e4T + 3.07e9T^{2} \)
83 \( 1 - 1.16e5iT - 3.93e9T^{2} \)
89 \( 1 + 9.93e4T + 5.58e9T^{2} \)
97 \( 1 + 6.29e4iT - 8.58e9T^{2} \)
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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

−11.01277752932689294008730999656, −9.989922800367691932897444498924, −9.288497811193936242632449100717, −8.092377294253905412302040529598, −6.81824051950181190872739160780, −6.52020878975219719257283815953, −4.65663529670992930553838686583, −3.91569259936469693538117750794, −2.63158289437345133413806386232, −1.06641592633823178225468432004, 0.810429624391299696056941661171, 1.62564952253153195137308559297, 3.60641523496034296960157088691, 4.42515647194237301215344724359, 5.68807772198855567401095538230, 6.84848423394397356921355122645, 7.77991800332152975760660809697, 8.727911070613679395513131531260, 9.599436713119141278944011012932, 10.59212273021381712845086500208

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