Properties

Degree $2$
Conductor $720$
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

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.948 + 0.317i$
Motivic weight: \(5\)
Character: $\chi_{720} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :5/2),\ -0.948 + 0.317i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.138196341\)
\(L(\frac12)\) \(\approx\) \(1.138196341\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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