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Properties

Label 2-1080-40.29-c1-0-45
Degree $2$
Conductor $1080$
Sign $0.678 - 0.734i$
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  + (−0.763 + 1.19i)2-s + (−0.835 − 1.81i)4-s + (1.27 − 1.83i)5-s + 1.23i·7-s + (2.80 + 0.391i)8-s + (1.21 + 2.91i)10-s + 4.64i·11-s + 4.85·13-s + (−1.46 − 0.940i)14-s + (−2.60 + 3.03i)16-s + 0.206i·17-s − 7.40i·19-s + (−4.40 − 0.782i)20-s + (−5.52 − 3.54i)22-s + 1.04i·23-s + ⋯
L(s)  = 1  + (−0.539 + 0.841i)2-s + (−0.417 − 0.908i)4-s + (0.570 − 0.821i)5-s + 0.465i·7-s + (0.990 + 0.138i)8-s + (0.383 + 0.923i)10-s + 1.39i·11-s + 1.34·13-s + (−0.392 − 0.251i)14-s + (−0.651 + 0.759i)16-s + 0.0500i·17-s − 1.69i·19-s + (−0.984 − 0.174i)20-s + (−1.17 − 0.754i)22-s + 0.218i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.678 - 0.734i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ 0.678 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.379677313\)
\(L(\frac12)\) \(\approx\) \(1.379677313\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.763 - 1.19i)T \)
3 \( 1 \)
5 \( 1 + (-1.27 + 1.83i)T \)
good7 \( 1 - 1.23iT - 7T^{2} \)
11 \( 1 - 4.64iT - 11T^{2} \)
13 \( 1 - 4.85T + 13T^{2} \)
17 \( 1 - 0.206iT - 17T^{2} \)
19 \( 1 + 7.40iT - 19T^{2} \)
23 \( 1 - 1.04iT - 23T^{2} \)
29 \( 1 - 6.52iT - 29T^{2} \)
31 \( 1 + 6.41T + 31T^{2} \)
37 \( 1 - 8.11T + 37T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 + 2.55T + 43T^{2} \)
47 \( 1 - 4.41iT - 47T^{2} \)
53 \( 1 - 3.80T + 53T^{2} \)
59 \( 1 - 5.04iT - 59T^{2} \)
61 \( 1 + 7.09iT - 61T^{2} \)
67 \( 1 - 13.9T + 67T^{2} \)
71 \( 1 + 9.12T + 71T^{2} \)
73 \( 1 - 8.71iT - 73T^{2} \)
79 \( 1 - 6.11T + 79T^{2} \)
83 \( 1 - 17.3T + 83T^{2} \)
89 \( 1 + 7.15T + 89T^{2} \)
97 \( 1 - 5.82iT - 97T^{2} \)
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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

−9.494940814586482911411037050479, −9.207726364661633820308623625258, −8.488908825531716958717438039211, −7.48818166606138938311501037110, −6.66631759934145727153726535491, −5.77621631398349805980081506404, −5.02776142084519674762961188990, −4.19572337887210805389481927681, −2.26038627807539037808346094844, −1.09757803766409354003279897486, 0.983759902139575316908129899741, 2.27020839627533127264621301545, 3.47529289382883617636395769615, 3.94003784080303050788780785950, 5.71789167239764635175733961254, 6.30023294792416116528924144383, 7.55899943501082939499307809635, 8.213779309649985059060300080713, 9.087441907908230100006498606768, 9.921499043676045704605281393797

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