Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $-0.983 + 0.178i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 14i·3-s + (−55 + 10i)5-s + 158i·7-s + 47·9-s + 148·11-s + 684i·13-s + (−140 − 770i)15-s + 2.04e3i·17-s + 2.22e3·19-s − 2.21e3·21-s + 1.24e3i·23-s + (2.92e3 − 1.10e3i)25-s + 4.06e3i·27-s − 270·29-s − 2.04e3·31-s + ⋯
L(s)  = 1  + 0.898i·3-s + (−0.983 + 0.178i)5-s + 1.21i·7-s + 0.193·9-s + 0.368·11-s + 1.12i·13-s + (−0.160 − 0.883i)15-s + 1.71i·17-s + 1.41·19-s − 1.09·21-s + 0.491i·23-s + (0.936 − 0.352i)25-s + 1.07i·27-s − 0.0596·29-s − 0.382·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $-0.983 + 0.178i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (129, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ -0.983 + 0.178i)\)
\(L(3)\)  \(\approx\)  \(1.547947274\)
\(L(\frac12)\)  \(\approx\)  \(1.547947274\)
\(L(\frac{7}{2})\)   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 \( 1 + (55 - 10i)T \)
good3 \( 1 - 14iT - 243T^{2} \)
7 \( 1 - 158iT - 1.68e4T^{2} \)
11 \( 1 - 148T + 1.61e5T^{2} \)
13 \( 1 - 684iT - 3.71e5T^{2} \)
17 \( 1 - 2.04e3iT - 1.41e6T^{2} \)
19 \( 1 - 2.22e3T + 2.47e6T^{2} \)
23 \( 1 - 1.24e3iT - 6.43e6T^{2} \)
29 \( 1 + 270T + 2.05e7T^{2} \)
31 \( 1 + 2.04e3T + 2.86e7T^{2} \)
37 \( 1 - 4.37e3iT - 6.93e7T^{2} \)
41 \( 1 + 2.39e3T + 1.15e8T^{2} \)
43 \( 1 - 2.29e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.06e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.96e3iT - 4.18e8T^{2} \)
59 \( 1 + 3.97e4T + 7.14e8T^{2} \)
61 \( 1 - 4.22e4T + 8.44e8T^{2} \)
67 \( 1 + 3.20e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.24e3T + 1.80e9T^{2} \)
73 \( 1 + 3.01e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.52e4T + 3.07e9T^{2} \)
83 \( 1 + 2.78e4iT - 3.93e9T^{2} \)
89 \( 1 - 8.52e4T + 5.58e9T^{2} \)
97 \( 1 + 9.72e4iT - 8.58e9T^{2} \)
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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

−11.40152723005750657822842957554, −10.34757264904187364532269775074, −9.341510265879589143710613731559, −8.684629844601267694369339341083, −7.56174395117571142419891896144, −6.39042117936015941783695381724, −5.18043803815000671230890681206, −4.11918762105717064503094414208, −3.31417070250116321552865347618, −1.64816353048125030239897637389, 0.50331916655101393402041627034, 1.06214537871068794523823957956, 2.96737626266581221367272840611, 4.09000221554575593486135417789, 5.22660877959376333887606380084, 6.85627125866067399252101557287, 7.41206681584715330809595035596, 7.976905079686492485291398375542, 9.362097390708126835512903973996, 10.40798133918283463611814069135

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