Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $0.836 + 0.548i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 24.1i·3-s + (−46.7 − 30.6i)5-s + 179. i·7-s − 339.·9-s − 653.·11-s + 284. i·13-s + (−740. + 1.12e3i)15-s − 383. i·17-s + 2.56e3·19-s + 4.34e3·21-s − 948. i·23-s + (1.24e3 + 2.86e3i)25-s + 2.33e3i·27-s − 1.52e3·29-s − 3.10e3·31-s + ⋯
L(s)  = 1  − 1.54i·3-s + (−0.836 − 0.548i)5-s + 1.38i·7-s − 1.39·9-s − 1.62·11-s + 0.467i·13-s + (−0.849 + 1.29i)15-s − 0.321i·17-s + 1.62·19-s + 2.14·21-s − 0.373i·23-s + (0.398 + 0.917i)25-s + 0.615i·27-s − 0.336·29-s − 0.580·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $0.836 + 0.548i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (129, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 0.836 + 0.548i)\)
\(L(3)\)  \(\approx\)  \(1.161262448\)
\(L(\frac12)\)  \(\approx\)  \(1.161262448\)
\(L(\frac{7}{2})\)   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 \( 1 + (46.7 + 30.6i)T \)
good3 \( 1 + 24.1iT - 243T^{2} \)
7 \( 1 - 179. iT - 1.68e4T^{2} \)
11 \( 1 + 653.T + 1.61e5T^{2} \)
13 \( 1 - 284. iT - 3.71e5T^{2} \)
17 \( 1 + 383. iT - 1.41e6T^{2} \)
19 \( 1 - 2.56e3T + 2.47e6T^{2} \)
23 \( 1 + 948. iT - 6.43e6T^{2} \)
29 \( 1 + 1.52e3T + 2.05e7T^{2} \)
31 \( 1 + 3.10e3T + 2.86e7T^{2} \)
37 \( 1 - 9.99e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.51e4T + 1.15e8T^{2} \)
43 \( 1 + 1.75e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.47e4iT - 2.29e8T^{2} \)
53 \( 1 + 8.70e3iT - 4.18e8T^{2} \)
59 \( 1 - 1.26e4T + 7.14e8T^{2} \)
61 \( 1 + 4.30e4T + 8.44e8T^{2} \)
67 \( 1 + 2.61e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.62e4T + 1.80e9T^{2} \)
73 \( 1 - 5.13e4iT - 2.07e9T^{2} \)
79 \( 1 - 3.93e4T + 3.07e9T^{2} \)
83 \( 1 - 6.95e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.30e4T + 5.58e9T^{2} \)
97 \( 1 + 2.62e4iT - 8.58e9T^{2} \)
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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

−11.15876952573403462470509368269, −9.517995176521661912332012881629, −8.487854338137981655923209997055, −7.83825862130286452789691270869, −7.09567307872275444431064484220, −5.75891997683435960771017377518, −5.01866794829273094048530039371, −3.04641866355727704039653927270, −2.07884099517618706255970374519, −0.70937336679334322433829747808, 0.47902289191581290682935811742, 2.99006423371546851500093409344, 3.73523338293967947581854227992, 4.65213470164039447468350705394, 5.64283446506670864891004842942, 7.46414682844236597571546950154, 7.76817410595285541313024298717, 9.290865834239280826977931253036, 10.23619659110538056178906680714, 10.72694261995924987206647105530

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