Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $-0.965 - 0.258i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 20.6i·3-s − 25i·5-s + 93.5·7-s − 182.·9-s − 659. i·11-s + 183. i·13-s − 515.·15-s − 467.·17-s − 225. i·19-s − 1.92e3i·21-s − 1.10e3·23-s − 625·25-s − 1.24e3i·27-s − 1.73e3i·29-s + 7.27e3·31-s + ⋯
L(s)  = 1  − 1.32i·3-s − 0.447i·5-s + 0.721·7-s − 0.751·9-s − 1.64i·11-s + 0.301i·13-s − 0.591·15-s − 0.391·17-s − 0.143i·19-s − 0.954i·21-s − 0.437·23-s − 0.200·25-s − 0.329i·27-s − 0.383i·29-s + 1.35·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $-0.965 - 0.258i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (161, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ -0.965 - 0.258i)\)
\(L(3)\)  \(\approx\)  \(1.643961073\)
\(L(\frac12)\)  \(\approx\)  \(1.643961073\)
\(L(\frac{7}{2})\)   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 \( 1 + 25iT \)
good3 \( 1 + 20.6iT - 243T^{2} \)
7 \( 1 - 93.5T + 1.68e4T^{2} \)
11 \( 1 + 659. iT - 1.61e5T^{2} \)
13 \( 1 - 183. iT - 3.71e5T^{2} \)
17 \( 1 + 467.T + 1.41e6T^{2} \)
19 \( 1 + 225. iT - 2.47e6T^{2} \)
23 \( 1 + 1.10e3T + 6.43e6T^{2} \)
29 \( 1 + 1.73e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.27e3T + 2.86e7T^{2} \)
37 \( 1 + 1.42e4iT - 6.93e7T^{2} \)
41 \( 1 + 4.92e3T + 1.15e8T^{2} \)
43 \( 1 + 1.23e4iT - 1.47e8T^{2} \)
47 \( 1 - 619.T + 2.29e8T^{2} \)
53 \( 1 - 1.69e4iT - 4.18e8T^{2} \)
59 \( 1 - 5.07e4iT - 7.14e8T^{2} \)
61 \( 1 + 2.06e4iT - 8.44e8T^{2} \)
67 \( 1 - 4.19e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.46e4T + 1.80e9T^{2} \)
73 \( 1 + 2.52e4T + 2.07e9T^{2} \)
79 \( 1 + 2.90e4T + 3.07e9T^{2} \)
83 \( 1 - 9.26e4iT - 3.93e9T^{2} \)
89 \( 1 - 5.46e4T + 5.58e9T^{2} \)
97 \( 1 - 4.55e4T + 8.58e9T^{2} \)
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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

−10.49804657963446228097010398282, −8.964449375895424474748723157535, −8.306742524470229932014503702185, −7.50265892999537329286961849192, −6.39028545611480904013043074918, −5.55559362304644580752451978724, −4.15324052023889329068449898395, −2.54998857417611203916032150702, −1.36821672993161962648371791319, −0.43638603285141010746659671203, 1.76546795500751770666797402947, 3.18099830583243793210980006241, 4.51013445575230626740086615285, 4.88534242081382501602178345161, 6.41475804734834724867298757525, 7.56921409308401802361050868275, 8.591334589477957300006163286634, 9.894160400569350318767991063191, 10.06598032643499962502249399298, 11.15006422051646612432388032475

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