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Properties

Label 2-1080-5.4-c3-0-25
Degree $2$
Conductor $1080$
Sign $0.999 - 0.0136i$
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  + (−11.1 + 0.152i)5-s − 26.3i·7-s + 18.4·11-s + 53.0i·13-s + 11.1i·17-s − 99.3·19-s + 144. i·23-s + (124. − 3.41i)25-s + 83.2·29-s − 313.·31-s + (4.02 + 294. i)35-s − 193. i·37-s + 158.·41-s + 180. i·43-s − 200. i·47-s + ⋯
L(s)  = 1  + (−0.999 + 0.0136i)5-s − 1.42i·7-s + 0.504·11-s + 1.13i·13-s + 0.159i·17-s − 1.20·19-s + 1.31i·23-s + (0.999 − 0.0273i)25-s + 0.532·29-s − 1.81·31-s + (0.0194 + 1.42i)35-s − 0.859i·37-s + 0.602·41-s + 0.641i·43-s − 0.622i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0136i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.999 - 0.0136i$
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: \((2,\ 1080,\ (\ :3/2),\ 0.999 - 0.0136i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.353068779\)
\(L(\frac12)\) \(\approx\) \(1.353068779\)
\(L(\frac{5}{2})\) 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 \( 1 + (11.1 - 0.152i)T \)
good7 \( 1 + 26.3iT - 343T^{2} \)
11 \( 1 - 18.4T + 1.33e3T^{2} \)
13 \( 1 - 53.0iT - 2.19e3T^{2} \)
17 \( 1 - 11.1iT - 4.91e3T^{2} \)
19 \( 1 + 99.3T + 6.85e3T^{2} \)
23 \( 1 - 144. iT - 1.21e4T^{2} \)
29 \( 1 - 83.2T + 2.43e4T^{2} \)
31 \( 1 + 313.T + 2.97e4T^{2} \)
37 \( 1 + 193. iT - 5.06e4T^{2} \)
41 \( 1 - 158.T + 6.89e4T^{2} \)
43 \( 1 - 180. iT - 7.95e4T^{2} \)
47 \( 1 + 200. iT - 1.03e5T^{2} \)
53 \( 1 + 252. iT - 1.48e5T^{2} \)
59 \( 1 - 117.T + 2.05e5T^{2} \)
61 \( 1 - 643.T + 2.26e5T^{2} \)
67 \( 1 + 661. iT - 3.00e5T^{2} \)
71 \( 1 - 707.T + 3.57e5T^{2} \)
73 \( 1 - 1.07e3iT - 3.89e5T^{2} \)
79 \( 1 - 338.T + 4.93e5T^{2} \)
83 \( 1 + 814. iT - 5.71e5T^{2} \)
89 \( 1 - 868.T + 7.04e5T^{2} \)
97 \( 1 + 566. iT - 9.12e5T^{2} \)
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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

−9.445354898712432311308210534019, −8.668968487614367407771045988080, −7.66597516003063104345345045994, −7.11413955243180639215059203909, −6.38865230558726895620772419402, −4.95210136168577766588690527979, −3.89766679557536291092106835446, −3.75323149861124182127796132748, −1.90596795356183890476993076226, −0.66084412133745497133261237159, 0.54891755420833739581093410124, 2.21599311196496246826699226510, 3.15405677043744660064795199855, 4.22468387312410631159371831947, 5.19697255004472466847094393625, 6.09289446121866422211731674455, 6.97990819801368754258090985070, 8.083856769510683836184224802062, 8.576250032495645907069455985940, 9.268551529225492589660143385704

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