Properties

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

Origins

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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

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $-0.769 - 0.638i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ -0.769 - 0.638i)\)
\(L(3)\)  \(\approx\)  \(0.9743192407\)
\(L(\frac12)\)  \(\approx\)  \(0.9743192407\)
\(L(\frac{7}{2})\)   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 \( 1 + 5.52e4iT - 8.58e9T^{2} \)
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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

−10.92763892801175985863577261705, −10.27818696233317601171952918432, −9.338159665055942583842525424685, −8.259287785559109665954251223047, −7.44750138317795342300794056464, −6.35372199856045652145558636727, −5.34035233239607324267238164557, −3.61066367278147927773417595419, −2.98636406741952689599501648819, −1.58654139492098372935389617169, 0.20456165174307449592924854723, 2.21283541180158623322246963329, 2.54874031531936460982580827212, 4.60578311111578956154631594910, 5.17913659603226942378147800594, 6.56842446098096208490859073006, 7.75452749570601267896234542541, 8.661161665896328387117933375925, 9.594930726164669569364421004149, 9.856033185778256766561393806241

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