Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 5 $
Sign $-0.983 + 0.178i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−55 + 10i)5-s − 158i·7-s − 148·11-s − 684i·13-s − 2.04e3i·17-s + 2.22e3·19-s + 1.24e3i·23-s + (2.92e3 − 1.10e3i)25-s − 270·29-s + 2.04e3·31-s + (1.58e3 + 8.69e3i)35-s − 4.37e3i·37-s + 2.39e3·41-s + 2.29e3i·43-s − 1.06e4i·47-s + ⋯
L(s)  = 1  + (−0.983 + 0.178i)5-s − 1.21i·7-s − 0.368·11-s − 1.12i·13-s − 1.71i·17-s + 1.41·19-s + 0.491i·23-s + (0.936 − 0.352i)25-s − 0.0596·29-s + 0.382·31-s + (0.218 + 1.19i)35-s − 0.525i·37-s + 0.222·41-s + 0.189i·43-s − 0.705i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
\( \varepsilon \)  =  $-0.983 + 0.178i$
motivic weight  =  \(5\)
character  :  $\chi_{720} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 720,\ (\ :5/2),\ -0.983 + 0.178i)\)
\(L(3)\)  \(\approx\)  \(1.123461028\)
\(L(\frac12)\)  \(\approx\)  \(1.123461028\)
\(L(\frac{7}{2})\)   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 \( 1 + (55 - 10i)T \)
good7 \( 1 + 158iT - 1.68e4T^{2} \)
11 \( 1 + 148T + 1.61e5T^{2} \)
13 \( 1 + 684iT - 3.71e5T^{2} \)
17 \( 1 + 2.04e3iT - 1.41e6T^{2} \)
19 \( 1 - 2.22e3T + 2.47e6T^{2} \)
23 \( 1 - 1.24e3iT - 6.43e6T^{2} \)
29 \( 1 + 270T + 2.05e7T^{2} \)
31 \( 1 - 2.04e3T + 2.86e7T^{2} \)
37 \( 1 + 4.37e3iT - 6.93e7T^{2} \)
41 \( 1 - 2.39e3T + 1.15e8T^{2} \)
43 \( 1 - 2.29e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.06e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.96e3iT - 4.18e8T^{2} \)
59 \( 1 - 3.97e4T + 7.14e8T^{2} \)
61 \( 1 + 4.22e4T + 8.44e8T^{2} \)
67 \( 1 + 3.20e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.24e3T + 1.80e9T^{2} \)
73 \( 1 + 3.01e4iT - 2.07e9T^{2} \)
79 \( 1 - 3.52e4T + 3.07e9T^{2} \)
83 \( 1 - 2.78e4iT - 3.93e9T^{2} \)
89 \( 1 + 8.52e4T + 5.58e9T^{2} \)
97 \( 1 + 9.72e4iT - 8.58e9T^{2} \)
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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

−9.355668036513258762201909828199, −8.077400479934443942902778531018, −7.47366335870589045784942961799, −7.00001335816922293298812854567, −5.48695229653567532681706656070, −4.65162063057916416811697579632, −3.55316429666574274638435600366, −2.86044218469418759283312630677, −0.940046955472601381627138948055, −0.30352029737504893863687685317, 1.28083330032161261943830772669, 2.50743321589217937955076141410, 3.62508818867320182130947131949, 4.59151064934415568329109083736, 5.58798640295237629456379136168, 6.52255924616614531058194645684, 7.58788739672034986071780418237, 8.423186770586250797940167920172, 8.985464609204727567999621514017, 9.997237747398641560083973834876

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