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  + 55.9·5-s + 245. i·7-s − 243·9-s − 802i·11-s − 245.·13-s − 1.91e3i·19-s − 3.33e3i·23-s + 3.12e3·25-s + 1.37e4i·35-s + 1.58e4·37-s + 1.52e4·41-s − 1.35e4·45-s + 1.53e4i·47-s − 4.36e4·49-s + 1.68e4·53-s + ⋯
L(s)  = 1  + 0.999·5-s + 1.89i·7-s − 9-s − 1.99i·11-s − 0.403·13-s − 1.21i·19-s − 1.31i·23-s + 25-s + 1.89i·35-s + 1.90·37-s + 1.41·41-s − 0.999·45-s + 1.01i·47-s − 2.59·49-s + 0.823·53-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  :  $\chi_{320} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 0.707 + 0.707i)\)
\(L(3)\)  \(\approx\)  \(2.006085520\)
\(L(\frac12)\)  \(\approx\)  \(2.006085520\)
\(L(\frac{7}{2})\)   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 \( 1 - 55.9T \)
good3 \( 1 + 243T^{2} \)
7 \( 1 - 245. iT - 1.68e4T^{2} \)
11 \( 1 + 802iT - 1.61e5T^{2} \)
13 \( 1 + 245.T + 3.71e5T^{2} \)
17 \( 1 - 1.41e6T^{2} \)
19 \( 1 + 1.91e3iT - 2.47e6T^{2} \)
23 \( 1 + 3.33e3iT - 6.43e6T^{2} \)
29 \( 1 - 2.05e7T^{2} \)
31 \( 1 + 2.86e7T^{2} \)
37 \( 1 - 1.58e4T + 6.93e7T^{2} \)
41 \( 1 - 1.52e4T + 1.15e8T^{2} \)
43 \( 1 + 1.47e8T^{2} \)
47 \( 1 - 1.53e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.68e4T + 4.18e8T^{2} \)
59 \( 1 + 2.79e4iT - 7.14e8T^{2} \)
61 \( 1 - 8.44e8T^{2} \)
67 \( 1 + 1.35e9T^{2} \)
71 \( 1 + 1.80e9T^{2} \)
73 \( 1 - 2.07e9T^{2} \)
79 \( 1 + 3.07e9T^{2} \)
83 \( 1 + 3.93e9T^{2} \)
89 \( 1 - 1.28e5T + 5.58e9T^{2} \)
97 \( 1 - 8.58e9T^{2} \)
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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

−10.93483592500462354177732095138, −9.407332922491448835665708566608, −8.883143342167209417097288441311, −8.226305163190499810521362711124, −6.23810048339743478371387697424, −5.92976261812794858455445699862, −5.01186322965047259224628202810, −2.82040731214383640035707393217, −2.52865423711880795406053004803, −0.57299733356626466473965749556, 1.11128989406410684344367236482, 2.27675507796219162943257077935, 3.84282787951566465870555092599, 4.86691942303921970454520533169, 6.05007141337034201631350484736, 7.19441764679067891009491935754, 7.77564199878126322771726769766, 9.399940886867519002305638119999, 9.978587801282841996252861914315, 10.65656454220006810757423457411

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