Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $-0.638 + 0.769i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 11.4·3-s + (32.3 + 45.6i)5-s + 121. i·7-s − 113.·9-s + 625. i·11-s − 924.·13-s + (−368. − 520. i)15-s + 1.32e3i·17-s − 1.13e3i·19-s − 1.38e3i·21-s − 3.05e3i·23-s + (−1.03e3 + 2.94e3i)25-s + 4.05e3·27-s + 8.23e3i·29-s − 1.98e3·31-s + ⋯
L(s)  = 1  − 0.731·3-s + (0.578 + 0.816i)5-s + 0.936i·7-s − 0.465·9-s + 1.55i·11-s − 1.51·13-s + (−0.422 − 0.596i)15-s + 1.11i·17-s − 0.721i·19-s − 0.684i·21-s − 1.20i·23-s + (−0.331 + 0.943i)25-s + 1.07·27-s + 1.81i·29-s − 0.371·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $-0.638 + 0.769i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ -0.638 + 0.769i)\)
\(L(3)\)  \(\approx\)  \(0.4860069401\)
\(L(\frac12)\)  \(\approx\)  \(0.4860069401\)
\(L(\frac{7}{2})\)   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 \( 1 + (-32.3 - 45.6i)T \)
good3 \( 1 + 11.4T + 243T^{2} \)
7 \( 1 - 121. iT - 1.68e4T^{2} \)
11 \( 1 - 625. iT - 1.61e5T^{2} \)
13 \( 1 + 924.T + 3.71e5T^{2} \)
17 \( 1 - 1.32e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.13e3iT - 2.47e6T^{2} \)
23 \( 1 + 3.05e3iT - 6.43e6T^{2} \)
29 \( 1 - 8.23e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.98e3T + 2.86e7T^{2} \)
37 \( 1 - 9.68e3T + 6.93e7T^{2} \)
41 \( 1 + 1.58e4T + 1.15e8T^{2} \)
43 \( 1 - 1.87e4T + 1.47e8T^{2} \)
47 \( 1 + 5.16e3iT - 2.29e8T^{2} \)
53 \( 1 + 5.65e3T + 4.18e8T^{2} \)
59 \( 1 + 2.86e4iT - 7.14e8T^{2} \)
61 \( 1 + 2.41e3iT - 8.44e8T^{2} \)
67 \( 1 - 4.58e4T + 1.35e9T^{2} \)
71 \( 1 + 5.53e3T + 1.80e9T^{2} \)
73 \( 1 + 5.88e4iT - 2.07e9T^{2} \)
79 \( 1 + 7.93e4T + 3.07e9T^{2} \)
83 \( 1 + 1.02e5T + 3.93e9T^{2} \)
89 \( 1 + 9.52e4T + 5.58e9T^{2} \)
97 \( 1 + 5.52e4iT - 8.58e9T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.34296774682421801908447954940, −10.47168899972873507731678540055, −9.715730188821835275460848242014, −8.715280380806245279005922658906, −7.26993494215730105252465839375, −6.54289780928873570642914830139, −5.50290030118147284843970352577, −4.69092371792360425170544313156, −2.79140053711952304798176724887, −1.98901206795813584989812841162, 0.16349092631900781744794517579, 0.957092580091985153326835410166, 2.70682597000327292377344861479, 4.24384797553780736436999991787, 5.39256440425535973012760912410, 5.94261269479500169056723936100, 7.29793717125327836667571694625, 8.276197349910688245243200968628, 9.450406880122519595454182785513, 10.14023898232953705435883975187

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