Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $0.933 + 0.357i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 28.5·3-s + (−5.81 + 55.5i)5-s − 92.3i·7-s + 571.·9-s − 541. i·11-s + 782.·13-s + (−165. + 1.58e3i)15-s − 1.55e3i·17-s − 484. i·19-s − 2.63e3i·21-s + 1.09e3i·23-s + (−3.05e3 − 646. i)25-s + 9.37e3·27-s − 597. i·29-s − 8.86e3·31-s + ⋯
L(s)  = 1  + 1.83·3-s + (−0.104 + 0.994i)5-s − 0.712i·7-s + 2.35·9-s − 1.34i·11-s + 1.28·13-s + (−0.190 + 1.82i)15-s − 1.30i·17-s − 0.307i·19-s − 1.30i·21-s + 0.432i·23-s + (−0.978 − 0.206i)25-s + 2.47·27-s − 0.131i·29-s − 1.65·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $0.933 + 0.357i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 0.933 + 0.357i)\)
\(L(3)\)  \(\approx\)  \(4.449182718\)
\(L(\frac12)\)  \(\approx\)  \(4.449182718\)
\(L(\frac{7}{2})\)   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 \( 1 + (5.81 - 55.5i)T \)
good3 \( 1 - 28.5T + 243T^{2} \)
7 \( 1 + 92.3iT - 1.68e4T^{2} \)
11 \( 1 + 541. iT - 1.61e5T^{2} \)
13 \( 1 - 782.T + 3.71e5T^{2} \)
17 \( 1 + 1.55e3iT - 1.41e6T^{2} \)
19 \( 1 + 484. iT - 2.47e6T^{2} \)
23 \( 1 - 1.09e3iT - 6.43e6T^{2} \)
29 \( 1 + 597. iT - 2.05e7T^{2} \)
31 \( 1 + 8.86e3T + 2.86e7T^{2} \)
37 \( 1 - 1.36e4T + 6.93e7T^{2} \)
41 \( 1 - 1.45e4T + 1.15e8T^{2} \)
43 \( 1 - 1.82e3T + 1.47e8T^{2} \)
47 \( 1 - 1.79e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.65e3T + 4.18e8T^{2} \)
59 \( 1 - 1.44e4iT - 7.14e8T^{2} \)
61 \( 1 + 2.45e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.68e4T + 1.35e9T^{2} \)
71 \( 1 + 4.80e4T + 1.80e9T^{2} \)
73 \( 1 - 6.06e4iT - 2.07e9T^{2} \)
79 \( 1 - 2.81e4T + 3.07e9T^{2} \)
83 \( 1 + 2.45e4T + 3.93e9T^{2} \)
89 \( 1 - 1.17e5T + 5.58e9T^{2} \)
97 \( 1 - 3.77e4iT - 8.58e9T^{2} \)
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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

−10.76773057347829149602018571040, −9.559298521187718021265602278915, −8.891242149272449688238729546903, −7.81726801503950987413970942185, −7.28578326276479880099462645516, −6.06723997894171510321315094874, −4.08648431596903245171667439305, −3.35439699359214894423777558755, −2.57837203431196001123498913481, −1.01307981331554902431355382655, 1.45644737573918076809809142163, 2.21820083244608168635939874310, 3.69340622144658444952959998486, 4.41623254759363890735470962921, 5.93633499897389256597849230513, 7.44307318777659801697589235617, 8.232078782099206244138350361673, 8.929563752112982245248557880794, 9.467774888278446811832065245939, 10.58415402367294274897073269328

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