Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $-0.707 + 0.707i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.72i·3-s + 25i·5-s − 65.1·7-s + 148.·9-s + 71.5i·11-s − 733. i·13-s + 243.·15-s − 716.·17-s + 2.33e3i·19-s + 633. i·21-s + 278.·23-s − 625·25-s − 3.80e3i·27-s − 2.95e3i·29-s + 6.04e3·31-s + ⋯
L(s)  = 1  − 0.623i·3-s + 0.447i·5-s − 0.502·7-s + 0.610·9-s + 0.178i·11-s − 1.20i·13-s + 0.279·15-s − 0.600·17-s + 1.48i·19-s + 0.313i·21-s + 0.109·23-s − 0.200·25-s − 1.00i·27-s − 0.652i·29-s + 1.13·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $-0.707 + 0.707i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (161, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ -0.707 + 0.707i)\)
\(L(3)\)  \(\approx\)  \(1.123371389\)
\(L(\frac12)\)  \(\approx\)  \(1.123371389\)
\(L(\frac{7}{2})\)   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 \( 1 + 9.72iT - 243T^{2} \)
7 \( 1 + 65.1T + 1.68e4T^{2} \)
11 \( 1 - 71.5iT - 1.61e5T^{2} \)
13 \( 1 + 733. iT - 3.71e5T^{2} \)
17 \( 1 + 716.T + 1.41e6T^{2} \)
19 \( 1 - 2.33e3iT - 2.47e6T^{2} \)
23 \( 1 - 278.T + 6.43e6T^{2} \)
29 \( 1 + 2.95e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.04e3T + 2.86e7T^{2} \)
37 \( 1 + 1.21e4iT - 6.93e7T^{2} \)
41 \( 1 - 3.62e3T + 1.15e8T^{2} \)
43 \( 1 + 230. iT - 1.47e8T^{2} \)
47 \( 1 + 1.39e4T + 2.29e8T^{2} \)
53 \( 1 + 2.24e4iT - 4.18e8T^{2} \)
59 \( 1 + 7.28e3iT - 7.14e8T^{2} \)
61 \( 1 + 1.94e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.39e3iT - 1.35e9T^{2} \)
71 \( 1 + 5.73e4T + 1.80e9T^{2} \)
73 \( 1 + 3.55e4T + 2.07e9T^{2} \)
79 \( 1 + 9.30e4T + 3.07e9T^{2} \)
83 \( 1 + 1.59e3iT - 3.93e9T^{2} \)
89 \( 1 - 2.02e3T + 5.58e9T^{2} \)
97 \( 1 + 1.20e5T + 8.58e9T^{2} \)
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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

−10.28336732000480891313385109743, −9.786303618067550857628712949113, −8.308845415894962328308832025572, −7.54357213713412200956756077214, −6.59136235650274552311584040926, −5.73813619386020357273302859634, −4.21849711060272298870995726461, −3.00438125201692724597275028982, −1.72860699056619363049462756238, −0.30206689174511111334824196239, 1.31116634097681017148781427033, 2.88629840241886848215338095833, 4.27949425133814525158761109936, 4.85437147147949406555086265547, 6.38499349748774901965652513679, 7.15687721920142806019577197295, 8.637223664726668143076012188382, 9.297974084310863900995796108533, 10.08379295973419454532797890053, 11.12592146664757670262576327370

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