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  + 5.98i·3-s + 25i·5-s − 86.0·7-s + 207.·9-s − 562. i·11-s + 659. i·13-s − 149.·15-s − 989.·17-s + 1.61e3i·19-s − 514. i·21-s + 2.93e3·23-s − 625·25-s + 2.69e3i·27-s + 5.65e3i·29-s − 6.87e3·31-s + ⋯
L(s)  = 1  + 0.383i·3-s + 0.447i·5-s − 0.663·7-s + 0.852·9-s − 1.40i·11-s + 1.08i·13-s − 0.171·15-s − 0.830·17-s + 1.02i·19-s − 0.254i·21-s + 1.15·23-s − 0.200·25-s + 0.710i·27-s + 1.24i·29-s − 1.28·31-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  =  \((2,\ 320,\ (\ :5/2),\ -0.965 + 0.258i)\)
\(L(3)\)  \(\approx\)  \(0.3450399730\)
\(L(\frac12)\)  \(\approx\)  \(0.3450399730\)
\(L(\frac{7}{2})\)   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 \( 1 - 5.98iT - 243T^{2} \)
7 \( 1 + 86.0T + 1.68e4T^{2} \)
11 \( 1 + 562. iT - 1.61e5T^{2} \)
13 \( 1 - 659. iT - 3.71e5T^{2} \)
17 \( 1 + 989.T + 1.41e6T^{2} \)
19 \( 1 - 1.61e3iT - 2.47e6T^{2} \)
23 \( 1 - 2.93e3T + 6.43e6T^{2} \)
29 \( 1 - 5.65e3iT - 2.05e7T^{2} \)
31 \( 1 + 6.87e3T + 2.86e7T^{2} \)
37 \( 1 + 1.08e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.80e4T + 1.15e8T^{2} \)
43 \( 1 + 1.48e4iT - 1.47e8T^{2} \)
47 \( 1 - 7.54e3T + 2.29e8T^{2} \)
53 \( 1 - 2.32e4iT - 4.18e8T^{2} \)
59 \( 1 - 9.08e3iT - 7.14e8T^{2} \)
61 \( 1 + 6.45e3iT - 8.44e8T^{2} \)
67 \( 1 + 6.20e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.95e4T + 1.80e9T^{2} \)
73 \( 1 - 2.15e3T + 2.07e9T^{2} \)
79 \( 1 + 2.56e4T + 3.07e9T^{2} \)
83 \( 1 + 5.48e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.15e5T + 5.58e9T^{2} \)
97 \( 1 + 1.14e5T + 8.58e9T^{2} \)
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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

−11.00515356120839754991193736555, −10.55873584859260675410613770763, −9.342128712785394648381216162936, −8.770614776301412775576834555294, −7.25988883530343163344829835711, −6.56686697067616857472281726850, −5.41728304701335426013676146673, −4.02323625134255001736821079544, −3.22786878150209700436200198378, −1.62716249252595876025730287151, 0.088946133704560300383513461476, 1.45012471293659643468251714355, 2.75647751806232953961439276016, 4.26965626784514995724573330258, 5.17246941426553150701036227553, 6.65171663270754852808353957779, 7.22258144887342003854458538450, 8.356197384277223526198417283006, 9.534555185541536158185879844987, 10.06744236152042739685388886233

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