Properties

Label 2-1080-1.1-c3-0-12
Degree $2$
Conductor $1080$
Sign $1$
Analytic cond. $63.7220$
Root an. cond. $7.98261$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s − 3.85·7-s + 42.2·11-s + 4.96·13-s − 25.8·17-s + 28.9·19-s − 191.·23-s + 25·25-s + 287.·29-s + 52.6·31-s + 19.2·35-s − 225.·37-s + 73.9·41-s − 275.·43-s − 192.·47-s − 328.·49-s + 275.·53-s − 211.·55-s + 497.·59-s + 44.0·61-s − 24.8·65-s + 761.·67-s + 264.·71-s + 728.·73-s − 163.·77-s + 664.·79-s + 1.49e3·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.208·7-s + 1.15·11-s + 0.105·13-s − 0.369·17-s + 0.350·19-s − 1.73·23-s + 0.200·25-s + 1.84·29-s + 0.305·31-s + 0.0931·35-s − 1.00·37-s + 0.281·41-s − 0.976·43-s − 0.596·47-s − 0.956·49-s + 0.713·53-s − 0.518·55-s + 1.09·59-s + 0.0924·61-s − 0.0473·65-s + 1.38·67-s + 0.441·71-s + 1.16·73-s − 0.241·77-s + 0.946·79-s + 1.97·83-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $1$
Analytic conductor: \(63.7220\)
Root analytic conductor: \(7.98261\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.814937102\)
\(L(\frac12)\) \(\approx\) \(1.814937102\)
\(L(\frac{5}{2})\) 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 \( 1 + 5T \)
good7 \( 1 + 3.85T + 343T^{2} \)
11 \( 1 - 42.2T + 1.33e3T^{2} \)
13 \( 1 - 4.96T + 2.19e3T^{2} \)
17 \( 1 + 25.8T + 4.91e3T^{2} \)
19 \( 1 - 28.9T + 6.85e3T^{2} \)
23 \( 1 + 191.T + 1.21e4T^{2} \)
29 \( 1 - 287.T + 2.43e4T^{2} \)
31 \( 1 - 52.6T + 2.97e4T^{2} \)
37 \( 1 + 225.T + 5.06e4T^{2} \)
41 \( 1 - 73.9T + 6.89e4T^{2} \)
43 \( 1 + 275.T + 7.95e4T^{2} \)
47 \( 1 + 192.T + 1.03e5T^{2} \)
53 \( 1 - 275.T + 1.48e5T^{2} \)
59 \( 1 - 497.T + 2.05e5T^{2} \)
61 \( 1 - 44.0T + 2.26e5T^{2} \)
67 \( 1 - 761.T + 3.00e5T^{2} \)
71 \( 1 - 264.T + 3.57e5T^{2} \)
73 \( 1 - 728.T + 3.89e5T^{2} \)
79 \( 1 - 664.T + 4.93e5T^{2} \)
83 \( 1 - 1.49e3T + 5.71e5T^{2} \)
89 \( 1 - 106.T + 7.04e5T^{2} \)
97 \( 1 + 924.T + 9.12e5T^{2} \)
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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

−9.568827125993691358247551415875, −8.554788506479603047017022812725, −8.016135375957208348443834637786, −6.77869701818573832443564971794, −6.37605361929033029909593553927, −5.11395257140662274878112368258, −4.12463576321131943517577708272, −3.36987185654270255202751829043, −2.01104205526282171519443229088, −0.71073809864372135639065978957, 0.71073809864372135639065978957, 2.01104205526282171519443229088, 3.36987185654270255202751829043, 4.12463576321131943517577708272, 5.11395257140662274878112368258, 6.37605361929033029909593553927, 6.77869701818573832443564971794, 8.016135375957208348443834637786, 8.554788506479603047017022812725, 9.568827125993691358247551415875

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