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Properties

Label 2-1080-9.7-c3-0-8
Degree $2$
Conductor $1080$
Sign $-0.989 + 0.145i$
Analytic cond. $63.7220$
Root an. cond. $7.98261$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.5 + 4.33i)5-s + (−17.3 + 30.0i)7-s + (−13.7 + 23.7i)11-s + (33.0 + 57.3i)13-s − 58.2·17-s + 106.·19-s + (10.3 + 17.9i)23-s + (−12.5 + 21.6i)25-s + (−95.7 + 165. i)29-s + (117. + 203. i)31-s − 173.·35-s − 63.4·37-s + (115. + 199. i)41-s + (202. − 351. i)43-s + (252. − 437. i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (−0.936 + 1.62i)7-s + (−0.376 + 0.652i)11-s + (0.705 + 1.22i)13-s − 0.830·17-s + 1.29·19-s + (0.0937 + 0.162i)23-s + (−0.100 + 0.173i)25-s + (−0.613 + 1.06i)29-s + (0.681 + 1.18i)31-s − 0.838·35-s − 0.281·37-s + (0.439 + 0.760i)41-s + (0.718 − 1.24i)43-s + (0.783 − 1.35i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.145i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.989 + 0.145i$
Analytic conductor: \(63.7220\)
Root analytic conductor: \(7.98261\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :3/2),\ -0.989 + 0.145i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.254719024\)
\(L(\frac12)\) \(\approx\) \(1.254719024\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.5 - 4.33i)T \)
good7 \( 1 + (17.3 - 30.0i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (13.7 - 23.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-33.0 - 57.3i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 58.2T + 4.91e3T^{2} \)
19 \( 1 - 106.T + 6.85e3T^{2} \)
23 \( 1 + (-10.3 - 17.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (95.7 - 165. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-117. - 203. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 63.4T + 5.06e4T^{2} \)
41 \( 1 + (-115. - 199. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-202. + 351. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-252. + 437. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 693.T + 1.48e5T^{2} \)
59 \( 1 + (218. + 379. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (287. - 498. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-351. - 608. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 238.T + 3.57e5T^{2} \)
73 \( 1 - 661.T + 3.89e5T^{2} \)
79 \( 1 + (288. - 499. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-625. + 1.08e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 402.T + 7.04e5T^{2} \)
97 \( 1 + (-409. + 708. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−9.699667054381893478774209120553, −9.154824377085213458528219169152, −8.574466925814882874461039918843, −7.19948299667819453756126237298, −6.58396881263063859754896264586, −5.74229475767109365046999745780, −4.91344285642718079732794497138, −3.54277145316023656090223400798, −2.64841020078929928465014025651, −1.70801675192981595812604754706, 0.34824048138637673589025697095, 1.02573074435147117870745669416, 2.83108532575864570532583092188, 3.67354689167919820124764839824, 4.58252349811680132929763977854, 5.82179869787264011695105924835, 6.39712343683377955259125931652, 7.62783084931477164365431409089, 7.966568781959159223327906410076, 9.327946325436140434522252747140

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