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Properties

Label 2-1080-5.4-c3-0-9
Degree $2$
Conductor $1080$
Sign $-0.610 - 0.792i$
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  + (6.82 + 8.85i)5-s − 20.2i·7-s − 35.6·11-s + 25.8i·13-s − 114. i·17-s + 29.6·19-s + 120. i·23-s + (−31.9 + 120. i)25-s − 270.·29-s + 198.·31-s + (179. − 138. i)35-s + 161. i·37-s − 201.·41-s + 444. i·43-s + 18.4i·47-s + ⋯
L(s)  = 1  + (0.610 + 0.792i)5-s − 1.09i·7-s − 0.976·11-s + 0.551i·13-s − 1.63i·17-s + 0.358·19-s + 1.09i·23-s + (−0.255 + 0.966i)25-s − 1.73·29-s + 1.15·31-s + (0.866 − 0.667i)35-s + 0.716i·37-s − 0.768·41-s + 1.57i·43-s + 0.0571i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 - 0.792i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.610 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.610 - 0.792i$
Analytic conductor: \(63.7220\)
Root analytic conductor: \(7.98261\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :3/2),\ -0.610 - 0.792i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9854031398\)
\(L(\frac12)\) \(\approx\) \(0.9854031398\)
\(L(\frac{5}{2})\) 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 \( 1 + (-6.82 - 8.85i)T \)
good7 \( 1 + 20.2iT - 343T^{2} \)
11 \( 1 + 35.6T + 1.33e3T^{2} \)
13 \( 1 - 25.8iT - 2.19e3T^{2} \)
17 \( 1 + 114. iT - 4.91e3T^{2} \)
19 \( 1 - 29.6T + 6.85e3T^{2} \)
23 \( 1 - 120. iT - 1.21e4T^{2} \)
29 \( 1 + 270.T + 2.43e4T^{2} \)
31 \( 1 - 198.T + 2.97e4T^{2} \)
37 \( 1 - 161. iT - 5.06e4T^{2} \)
41 \( 1 + 201.T + 6.89e4T^{2} \)
43 \( 1 - 444. iT - 7.95e4T^{2} \)
47 \( 1 - 18.4iT - 1.03e5T^{2} \)
53 \( 1 - 681. iT - 1.48e5T^{2} \)
59 \( 1 - 456.T + 2.05e5T^{2} \)
61 \( 1 + 612.T + 2.26e5T^{2} \)
67 \( 1 + 659. iT - 3.00e5T^{2} \)
71 \( 1 + 455.T + 3.57e5T^{2} \)
73 \( 1 - 178. iT - 3.89e5T^{2} \)
79 \( 1 - 586.T + 4.93e5T^{2} \)
83 \( 1 - 778. iT - 5.71e5T^{2} \)
89 \( 1 - 747.T + 7.04e5T^{2} \)
97 \( 1 + 1.17e3iT - 9.12e5T^{2} \)
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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

−9.742801442670050998695553716420, −9.317878759909673514233867750076, −7.78188411490785098843276707043, −7.36972468818772319194960457696, −6.55977556399787755302595382612, −5.51647232894018931475789091955, −4.64476771890322414180697281241, −3.42005682045191721733278273255, −2.57811113861262238802634190240, −1.26400208222149389167190480052, 0.23260993263374924580734218762, 1.77033240499474822690924442082, 2.57700432574323700173240152960, 3.91612827458590685074733198632, 5.19753771166993975065591940400, 5.59123779563074365359024701062, 6.44604711571361285000640167939, 7.81273423387184619420859275917, 8.495187000761169242049948135895, 9.036522271191346407557887281608

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