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Properties

Label 2-1080-5.4-c3-0-1
Degree $2$
Conductor $1080$
Sign $0.0285 - 0.999i$
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  + (0.319 − 11.1i)5-s − 17.5i·7-s − 59.7·11-s − 35.3i·13-s + 57.2i·17-s − 51.0·19-s + 27.5i·23-s + (−124. − 7.13i)25-s − 76.2·29-s − 142.·31-s + (−196. − 5.61i)35-s + 73.3i·37-s + 464.·41-s + 212. i·43-s − 166. i·47-s + ⋯
L(s)  = 1  + (0.0285 − 0.999i)5-s − 0.948i·7-s − 1.63·11-s − 0.754i·13-s + 0.816i·17-s − 0.616·19-s + 0.250i·23-s + (−0.998 − 0.0571i)25-s − 0.488·29-s − 0.828·31-s + (−0.948 − 0.0271i)35-s + 0.325i·37-s + 1.76·41-s + 0.755i·43-s − 0.517i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0285 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0285 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.0285 - 0.999i$
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: \((2,\ 1080,\ (\ :3/2),\ 0.0285 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3005102058\)
\(L(\frac12)\) \(\approx\) \(0.3005102058\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.319 + 11.1i)T \)
good7 \( 1 + 17.5iT - 343T^{2} \)
11 \( 1 + 59.7T + 1.33e3T^{2} \)
13 \( 1 + 35.3iT - 2.19e3T^{2} \)
17 \( 1 - 57.2iT - 4.91e3T^{2} \)
19 \( 1 + 51.0T + 6.85e3T^{2} \)
23 \( 1 - 27.5iT - 1.21e4T^{2} \)
29 \( 1 + 76.2T + 2.43e4T^{2} \)
31 \( 1 + 142.T + 2.97e4T^{2} \)
37 \( 1 - 73.3iT - 5.06e4T^{2} \)
41 \( 1 - 464.T + 6.89e4T^{2} \)
43 \( 1 - 212. iT - 7.95e4T^{2} \)
47 \( 1 + 166. iT - 1.03e5T^{2} \)
53 \( 1 - 744. iT - 1.48e5T^{2} \)
59 \( 1 + 18.9T + 2.05e5T^{2} \)
61 \( 1 - 83.2T + 2.26e5T^{2} \)
67 \( 1 + 644. iT - 3.00e5T^{2} \)
71 \( 1 - 1.03e3T + 3.57e5T^{2} \)
73 \( 1 + 1.08e3iT - 3.89e5T^{2} \)
79 \( 1 - 512.T + 4.93e5T^{2} \)
83 \( 1 - 69.4iT - 5.71e5T^{2} \)
89 \( 1 + 864.T + 7.04e5T^{2} \)
97 \( 1 - 686. iT - 9.12e5T^{2} \)
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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

−9.741927395021154997568414593636, −8.829743058857719836124168871702, −7.79989650984706183645309322646, −7.66166824783105922339844171331, −6.15738497938810321906379640904, −5.36247046231773956721546904485, −4.53478984597288013422824174701, −3.59295325852422364413574838225, −2.26321860964275180098664713873, −0.953848325497771021064440795807, 0.082260954006236153877393997651, 2.21714493708253549528479686821, 2.63075413622446773835622660345, 3.88856366935830185760641925116, 5.16342833318360664462089902902, 5.82406030085741272212099653880, 6.84903454205959845084033165966, 7.56184558559092443525367407837, 8.456003249976148377452212695064, 9.363504367965705625403389849123

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