Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $0.804 + 0.593i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 19.8i·3-s + (45 + 33.1i)5-s + 59.6i·7-s − 153·9-s − 252·11-s − 119. i·13-s + (660 − 895. i)15-s + 689. i·17-s + 220·19-s + 1.18e3·21-s − 2.43e3i·23-s + (924. + 2.98e3i)25-s − 1.79e3i·27-s + 6.93e3·29-s + 6.75e3·31-s + ⋯
L(s)  = 1  − 1.27i·3-s + (0.804 + 0.593i)5-s + 0.460i·7-s − 0.629·9-s − 0.627·11-s − 0.195i·13-s + (0.757 − 1.02i)15-s + 0.578i·17-s + 0.139·19-s + 0.587·21-s − 0.959i·23-s + (0.295 + 0.955i)25-s − 0.472i·27-s + 1.53·29-s + 1.26·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $0.804 + 0.593i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (129, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 0.804 + 0.593i)\)
\(L(3)\)  \(\approx\)  \(2.451800411\)
\(L(\frac12)\)  \(\approx\)  \(2.451800411\)
\(L(\frac{7}{2})\)   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 \( 1 + (-45 - 33.1i)T \)
good3 \( 1 + 19.8iT - 243T^{2} \)
7 \( 1 - 59.6iT - 1.68e4T^{2} \)
11 \( 1 + 252T + 1.61e5T^{2} \)
13 \( 1 + 119. iT - 3.71e5T^{2} \)
17 \( 1 - 689. iT - 1.41e6T^{2} \)
19 \( 1 - 220T + 2.47e6T^{2} \)
23 \( 1 + 2.43e3iT - 6.43e6T^{2} \)
29 \( 1 - 6.93e3T + 2.05e7T^{2} \)
31 \( 1 - 6.75e3T + 2.86e7T^{2} \)
37 \( 1 - 1.39e4iT - 6.93e7T^{2} \)
41 \( 1 + 198T + 1.15e8T^{2} \)
43 \( 1 + 417. iT - 1.47e8T^{2} \)
47 \( 1 - 1.05e4iT - 2.29e8T^{2} \)
53 \( 1 + 5.82e3iT - 4.18e8T^{2} \)
59 \( 1 - 2.46e4T + 7.14e8T^{2} \)
61 \( 1 - 5.69e3T + 8.44e8T^{2} \)
67 \( 1 + 4.36e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.33e4T + 1.80e9T^{2} \)
73 \( 1 + 7.09e4iT - 2.07e9T^{2} \)
79 \( 1 - 5.19e4T + 3.07e9T^{2} \)
83 \( 1 + 6.18e4iT - 3.93e9T^{2} \)
89 \( 1 + 9.99e3T + 5.58e9T^{2} \)
97 \( 1 - 1.01e5iT - 8.58e9T^{2} \)
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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

−10.61885727057638635489099094405, −9.936509329006115361329349307374, −8.566206397737891162007650538369, −7.81633255876696473312750953196, −6.60695617428181499303727716187, −6.20307045251558901401140122682, −4.90140162763634316618110025823, −2.92752251636657202497435596053, −2.13458075750337432354014031422, −0.906969510706608520532566786688, 0.875626916871380084664835081593, 2.53844537548554532048530836116, 3.93362606396282082009758789029, 4.87791178937101566643690811348, 5.61103525918812690149165152958, 6.99452933116971591506472427072, 8.322391349587799893879498164950, 9.279836798413377180412037199013, 9.985222696342128805290826051643, 10.54762364542483702189177605132

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