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Properties

Label 2-1080-8.5-c1-0-16
Degree $2$
Conductor $1080$
Sign $0.589 - 0.808i$
Analytic cond. $8.62384$
Root an. cond. $2.93663$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.34 + 0.436i)2-s + (1.61 − 1.17i)4-s + i·5-s − 3.61·7-s + (−1.66 + 2.28i)8-s + (−0.436 − 1.34i)10-s + 0.460i·11-s − 3.67i·13-s + (4.86 − 1.57i)14-s + (1.24 − 3.80i)16-s + 6.59·17-s − 3.13i·19-s + (1.17 + 1.61i)20-s + (−0.200 − 0.619i)22-s − 3.60·23-s + ⋯
L(s)  = 1  + (−0.951 + 0.308i)2-s + (0.809 − 0.586i)4-s + 0.447i·5-s − 1.36·7-s + (−0.589 + 0.808i)8-s + (−0.137 − 0.425i)10-s + 0.138i·11-s − 1.01i·13-s + (1.29 − 0.421i)14-s + (0.310 − 0.950i)16-s + 1.59·17-s − 0.719i·19-s + (0.262 + 0.362i)20-s + (−0.0428 − 0.132i)22-s − 0.751·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 - 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.589 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.589 - 0.808i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ 0.589 - 0.808i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8166837663\)
\(L(\frac12)\) \(\approx\) \(0.8166837663\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 - 0.436i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 + 3.61T + 7T^{2} \)
11 \( 1 - 0.460iT - 11T^{2} \)
13 \( 1 + 3.67iT - 13T^{2} \)
17 \( 1 - 6.59T + 17T^{2} \)
19 \( 1 + 3.13iT - 19T^{2} \)
23 \( 1 + 3.60T + 23T^{2} \)
29 \( 1 - 9.83iT - 29T^{2} \)
31 \( 1 - 9.34T + 31T^{2} \)
37 \( 1 - 6.66iT - 37T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 - 6.05iT - 43T^{2} \)
47 \( 1 + 6.61T + 47T^{2} \)
53 \( 1 + 3.27iT - 53T^{2} \)
59 \( 1 - 5.19iT - 59T^{2} \)
61 \( 1 + 6.80iT - 61T^{2} \)
67 \( 1 - 10.0iT - 67T^{2} \)
71 \( 1 - 2.98T + 71T^{2} \)
73 \( 1 - 2.27T + 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 - 6.88iT - 83T^{2} \)
89 \( 1 + 6.05T + 89T^{2} \)
97 \( 1 - 13.0T + 97T^{2} \)
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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

−9.998478332886543567056922733599, −9.357251215744685845867291889922, −8.265171848458687678768758283206, −7.59976017130628707685165729819, −6.66989243422820948615368755097, −6.10641161808854533176649782089, −5.12630075274087239953217049701, −3.33416555273615760389146492626, −2.75240455110413838290263820701, −0.946120852594128754792806616782, 0.66083414334482576422162600236, 2.14736855127287813848812758925, 3.33059659924851214908731746440, 4.17153326755324987577125370266, 5.90465965659732688638108561434, 6.35787905634467446693001557364, 7.51398226966087492480928010581, 8.142249113898538108466863989428, 9.156366392135136405745750164458, 9.805221936536078419925445256167

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