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Properties

Label 2-1080-8.5-c1-0-61
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

\(L(1)\) \(\approx\) \(2.079170898\)
\(L(\frac12)\) \(\approx\) \(2.079170898\)
\(L(\frac{3}{2})\) 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

−9.806151373754150210538235313044, −8.912413015747287039033817813491, −7.83683129105383094819039837330, −6.56035414271489608060755879082, −6.34209139262397116045795342112, −5.11936897313378103429548371542, −4.34320772068755016147202916702, −3.21359239697554705467920917519, −2.46879610804079299190440552752, −0.63509008168270703440120980076, 2.09940084958845189832158785402, 3.14189665596864352468671032583, 3.96189025549577705292095061427, 4.94638280466450443497910266219, 6.10966920380187099331352254461, 6.78771249286762989968233291696, 7.11836430080673981976863721177, 8.537985152451432522005534432450, 9.263761007835482170555203622559, 10.37226868931629465995916927387

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