Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 5 $
Sign $-0.948 - 0.317i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (53.0 + 17.7i)5-s + 188. i·7-s − 501.·11-s − 1.06e3i·13-s + 29.5i·17-s + 1.57e3·19-s + 1.29e3i·23-s + (2.49e3 + 1.88e3i)25-s − 3.58e3·29-s + 3.52e3·31-s + (−3.35e3 + 1.00e4i)35-s + 8.41e3i·37-s − 7.01e3·41-s + 2.26e4i·43-s − 3.50e3i·47-s + ⋯
L(s)  = 1  + (0.948 + 0.317i)5-s + 1.45i·7-s − 1.25·11-s − 1.74i·13-s + 0.0248i·17-s + 1.00·19-s + 0.510i·23-s + (0.798 + 0.602i)25-s − 0.791·29-s + 0.659·31-s + (−0.463 + 1.38i)35-s + 1.01i·37-s − 0.651·41-s + 1.87i·43-s − 0.231i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.317i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
\( \varepsilon \)  =  $-0.948 - 0.317i$
motivic weight  =  \(5\)
character  :  $\chi_{720} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 720,\ (\ :5/2),\ -0.948 - 0.317i)\)
\(L(3)\)  \(\approx\)  \(1.138196341\)
\(L(\frac12)\)  \(\approx\)  \(1.138196341\)
\(L(\frac{7}{2})\)   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 \( 1 + (-53.0 - 17.7i)T \)
good7 \( 1 - 188. iT - 1.68e4T^{2} \)
11 \( 1 + 501.T + 1.61e5T^{2} \)
13 \( 1 + 1.06e3iT - 3.71e5T^{2} \)
17 \( 1 - 29.5iT - 1.41e6T^{2} \)
19 \( 1 - 1.57e3T + 2.47e6T^{2} \)
23 \( 1 - 1.29e3iT - 6.43e6T^{2} \)
29 \( 1 + 3.58e3T + 2.05e7T^{2} \)
31 \( 1 - 3.52e3T + 2.86e7T^{2} \)
37 \( 1 - 8.41e3iT - 6.93e7T^{2} \)
41 \( 1 + 7.01e3T + 1.15e8T^{2} \)
43 \( 1 - 2.26e4iT - 1.47e8T^{2} \)
47 \( 1 + 3.50e3iT - 2.29e8T^{2} \)
53 \( 1 - 2.73e4iT - 4.18e8T^{2} \)
59 \( 1 + 7.92e3T + 7.14e8T^{2} \)
61 \( 1 + 7.02e3T + 8.44e8T^{2} \)
67 \( 1 + 1.76e4iT - 1.35e9T^{2} \)
71 \( 1 - 1.34e4T + 1.80e9T^{2} \)
73 \( 1 + 3.99e4iT - 2.07e9T^{2} \)
79 \( 1 + 9.33e4T + 3.07e9T^{2} \)
83 \( 1 - 5.84e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.39e4T + 5.58e9T^{2} \)
97 \( 1 + 1.10e5iT - 8.58e9T^{2} \)
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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

−9.979943380394942567915323417220, −9.319666731975254480169046148848, −8.259686572781381981065565044938, −7.61089391307246936319448055704, −6.23374320655105945572158237837, −5.50516293887539343982221290495, −5.10624140582134474995688137501, −2.98641798392962560008580654481, −2.73804949833829672232757428643, −1.38416421474668904496555765184, 0.22106664779076473625456166777, 1.39368699581079913726855152068, 2.39633392741632630567932788603, 3.80027660480860817256968660284, 4.73063168591573360327359633312, 5.59774254518515264310342046061, 6.79354132812600344602976107006, 7.33285876419298077946466711820, 8.457688763151813699194836016865, 9.421760270235857254935757879211

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