Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.06i·3-s + 25i·5-s − 250.·7-s + 217.·9-s − 94.5i·11-s − 887. i·13-s − 126.·15-s + 1.44e3·17-s − 611. i·19-s − 1.26e3i·21-s + 814.·23-s − 625·25-s + 2.33e3i·27-s + 5.29e3i·29-s + 3.61e3·31-s + ⋯
L(s)  = 1  + 0.325i·3-s + 0.447i·5-s − 1.93·7-s + 0.894·9-s − 0.235i·11-s − 1.45i·13-s − 0.145·15-s + 1.21·17-s − 0.388i·19-s − 0.627i·21-s + 0.320·23-s − 0.200·25-s + 0.615i·27-s + 1.16i·29-s + 0.675·31-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $0.258 - 0.965i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (161, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 0.258 - 0.965i)\)
\(L(3)\)  \(\approx\)  \(1.421807343\)
\(L(\frac12)\)  \(\approx\)  \(1.421807343\)
\(L(\frac{7}{2})\)   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 \( 1 - 5.06iT - 243T^{2} \)
7 \( 1 + 250.T + 1.68e4T^{2} \)
11 \( 1 + 94.5iT - 1.61e5T^{2} \)
13 \( 1 + 887. iT - 3.71e5T^{2} \)
17 \( 1 - 1.44e3T + 1.41e6T^{2} \)
19 \( 1 + 611. iT - 2.47e6T^{2} \)
23 \( 1 - 814.T + 6.43e6T^{2} \)
29 \( 1 - 5.29e3iT - 2.05e7T^{2} \)
31 \( 1 - 3.61e3T + 2.86e7T^{2} \)
37 \( 1 - 9.48e3iT - 6.93e7T^{2} \)
41 \( 1 + 9.91e3T + 1.15e8T^{2} \)
43 \( 1 - 2.02e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.07e4T + 2.29e8T^{2} \)
53 \( 1 + 1.30e4iT - 4.18e8T^{2} \)
59 \( 1 + 1.66e4iT - 7.14e8T^{2} \)
61 \( 1 + 715. iT - 8.44e8T^{2} \)
67 \( 1 - 2.19e4iT - 1.35e9T^{2} \)
71 \( 1 + 3.84e4T + 1.80e9T^{2} \)
73 \( 1 + 4.08e4T + 2.07e9T^{2} \)
79 \( 1 - 8.48e4T + 3.07e9T^{2} \)
83 \( 1 - 1.16e5iT - 3.93e9T^{2} \)
89 \( 1 - 1.21e5T + 5.58e9T^{2} \)
97 \( 1 - 2.15e4T + 8.58e9T^{2} \)
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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

−10.63124805204061588870124551850, −10.03791515341979631125323548499, −9.493357723932258886195610153641, −8.100138480338976116673504499773, −6.98796235907540620105859002873, −6.24180436859132184117268673720, −5.06309130820656883916680859066, −3.40684809176450250412468931746, −3.06856239524724361466792817603, −0.918845148119916677974166588374, 0.48326786052838275737283059647, 1.88243691056100744289611501858, 3.39466062514337089438868190899, 4.36920395024835558416678339896, 5.89922853603971574906878913357, 6.73373790074248527951490344797, 7.51443132956611071224770292549, 8.940343817088361720436401823268, 9.724591472078846928798826150597, 10.25091946821376593111490833373

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