Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 5 $
Sign $0.836 + 0.548i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−46.7 − 30.6i)5-s + 179. i·7-s − 653.·11-s − 284. i·13-s + 383. i·17-s − 2.56e3·19-s + 948. i·23-s + (1.24e3 + 2.86e3i)25-s − 1.52e3·29-s − 3.10e3·31-s + (5.51e3 − 8.40e3i)35-s − 9.99e3i·37-s − 1.51e4·41-s + 1.75e3i·43-s + 1.47e4i·47-s + ⋯
L(s)  = 1  + (−0.836 − 0.548i)5-s + 1.38i·7-s − 1.62·11-s − 0.467i·13-s + 0.321i·17-s − 1.62·19-s + 0.373i·23-s + (0.398 + 0.917i)25-s − 0.336·29-s − 0.580·31-s + (0.761 − 1.16i)35-s − 1.19i·37-s − 1.40·41-s + 0.144i·43-s + 0.974i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
\( \varepsilon \)  =  $0.836 + 0.548i$
motivic weight  =  \(5\)
character  :  $\chi_{720} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 720,\ (\ :5/2),\ 0.836 + 0.548i)\)
\(L(3)\)  \(\approx\)  \(0.5615320340\)
\(L(\frac12)\)  \(\approx\)  \(0.5615320340\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (46.7 + 30.6i)T \)
good7 \( 1 - 179. iT - 1.68e4T^{2} \)
11 \( 1 + 653.T + 1.61e5T^{2} \)
13 \( 1 + 284. iT - 3.71e5T^{2} \)
17 \( 1 - 383. iT - 1.41e6T^{2} \)
19 \( 1 + 2.56e3T + 2.47e6T^{2} \)
23 \( 1 - 948. iT - 6.43e6T^{2} \)
29 \( 1 + 1.52e3T + 2.05e7T^{2} \)
31 \( 1 + 3.10e3T + 2.86e7T^{2} \)
37 \( 1 + 9.99e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.51e4T + 1.15e8T^{2} \)
43 \( 1 - 1.75e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.47e4iT - 2.29e8T^{2} \)
53 \( 1 + 8.70e3iT - 4.18e8T^{2} \)
59 \( 1 - 1.26e4T + 7.14e8T^{2} \)
61 \( 1 - 4.30e4T + 8.44e8T^{2} \)
67 \( 1 - 2.61e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.62e4T + 1.80e9T^{2} \)
73 \( 1 - 5.13e4iT - 2.07e9T^{2} \)
79 \( 1 - 3.93e4T + 3.07e9T^{2} \)
83 \( 1 - 6.95e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.30e4T + 5.58e9T^{2} \)
97 \( 1 + 2.62e4iT - 8.58e9T^{2} \)
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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

−9.434080143602035628960332627511, −8.399572329402447046189893175078, −8.217712057228164103750720476519, −7.06609753252688608557316825256, −5.70229607137467158424781246345, −5.24213163003096622900014805578, −4.09250467650701867356110791737, −2.88043736509264931942810638882, −1.95594145045587337466972743682, −0.23594386061335684758237221877, 0.46430658500717263471593559093, 2.12246412202089530908351919487, 3.29460425961799727338291502189, 4.19740576142539849702784215618, 5.00995807957939437575856828763, 6.48730153622841701735832331129, 7.16799108037831178357346538656, 7.901551872396759529925908121171, 8.631080379949539319258030370278, 10.13776574160028253264505719635

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