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Properties

Label 2-1080-8.5-c1-0-39
Degree $2$
Conductor $1080$
Sign $0.994 + 0.107i$
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  + (0.662 + 1.24i)2-s + (−1.12 + 1.65i)4-s − i·5-s − 1.52·7-s + (−2.81 − 0.303i)8-s + (1.24 − 0.662i)10-s − 5.34i·11-s + 0.986i·13-s + (−1.01 − 1.90i)14-s + (−1.48 − 3.71i)16-s + 5.84·17-s − 7.86i·19-s + (1.65 + 1.12i)20-s + (6.68 − 3.54i)22-s + 4.41·23-s + ⋯
L(s)  = 1  + (0.468 + 0.883i)2-s + (−0.560 + 0.828i)4-s − 0.447i·5-s − 0.577·7-s + (−0.994 − 0.107i)8-s + (0.395 − 0.209i)10-s − 1.61i·11-s + 0.273i·13-s + (−0.270 − 0.510i)14-s + (−0.371 − 0.928i)16-s + 1.41·17-s − 1.80i·19-s + (0.370 + 0.250i)20-s + (1.42 − 0.755i)22-s + 0.921·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.994 + 0.107i$
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: \((2,\ 1080,\ (\ :1/2),\ 0.994 + 0.107i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.590002897\)
\(L(\frac12)\) \(\approx\) \(1.590002897\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.662 - 1.24i)T \)
3 \( 1 \)
5 \( 1 + iT \)
good7 \( 1 + 1.52T + 7T^{2} \)
11 \( 1 + 5.34iT - 11T^{2} \)
13 \( 1 - 0.986iT - 13T^{2} \)
17 \( 1 - 5.84T + 17T^{2} \)
19 \( 1 + 7.86iT - 19T^{2} \)
23 \( 1 - 4.41T + 23T^{2} \)
29 \( 1 - 9.95iT - 29T^{2} \)
31 \( 1 - 3.04T + 31T^{2} \)
37 \( 1 + 11.2iT - 37T^{2} \)
41 \( 1 + 8.75T + 41T^{2} \)
43 \( 1 - 3.01iT - 43T^{2} \)
47 \( 1 - 9.74T + 47T^{2} \)
53 \( 1 + 5.83iT - 53T^{2} \)
59 \( 1 + 5.85iT - 59T^{2} \)
61 \( 1 + 6.88iT - 61T^{2} \)
67 \( 1 + 3.97iT - 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + 6.83T + 73T^{2} \)
79 \( 1 + 2.07T + 79T^{2} \)
83 \( 1 - 4.30iT - 83T^{2} \)
89 \( 1 - 3.01T + 89T^{2} \)
97 \( 1 + 5.12T + 97T^{2} \)
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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

−9.454310772567820541603102889749, −8.924021922497444903569035611544, −8.232828480301933920646656259658, −7.18798127015325094336550469718, −6.50998270036026420871628089285, −5.50998029117906163335071390665, −4.96828555420584075809223052177, −3.59990778099170374455955343059, −2.99259499575175601144767967474, −0.66278881734308193325613815792, 1.40613008270371729858944006271, 2.63472113827779697018339666862, 3.54970474574898255763530409484, 4.46657338975569160501110708194, 5.52792264184380637179904664885, 6.32315164055160169424764217236, 7.36583880469287191695207795961, 8.283575200957308671050073938609, 9.582433740662162813189429298982, 10.09315188713165313806321571256

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