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Properties

Label 2-1080-40.29-c1-0-64
Degree $2$
Conductor $1080$
Sign $-0.914 - 0.405i$
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  + (−0.564 − 1.29i)2-s + (−1.36 + 1.46i)4-s + (−2.22 − 0.173i)5-s + 3.43i·7-s + (2.66 + 0.943i)8-s + (1.03 + 2.98i)10-s − 4.54i·11-s + 1.84·13-s + (4.45 − 1.93i)14-s + (−0.281 − 3.99i)16-s + 0.380i·17-s − 1.23i·19-s + (3.29 − 3.02i)20-s + (−5.89 + 2.56i)22-s + 5.35i·23-s + ⋯
L(s)  = 1  + (−0.398 − 0.917i)2-s + (−0.681 + 0.731i)4-s + (−0.996 − 0.0774i)5-s + 1.29i·7-s + (0.942 + 0.333i)8-s + (0.326 + 0.945i)10-s − 1.37i·11-s + 0.510·13-s + (1.18 − 0.517i)14-s + (−0.0702 − 0.997i)16-s + 0.0923i·17-s − 0.282i·19-s + (0.736 − 0.676i)20-s + (−1.25 + 0.546i)22-s + 1.11i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.914 - 0.405i$
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: \((2,\ 1080,\ (\ :1/2),\ -0.914 - 0.405i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1379541166\)
\(L(\frac12)\) \(\approx\) \(0.1379541166\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.564 + 1.29i)T \)
3 \( 1 \)
5 \( 1 + (2.22 + 0.173i)T \)
good7 \( 1 - 3.43iT - 7T^{2} \)
11 \( 1 + 4.54iT - 11T^{2} \)
13 \( 1 - 1.84T + 13T^{2} \)
17 \( 1 - 0.380iT - 17T^{2} \)
19 \( 1 + 1.23iT - 19T^{2} \)
23 \( 1 - 5.35iT - 23T^{2} \)
29 \( 1 + 3.17iT - 29T^{2} \)
31 \( 1 + 6.89T + 31T^{2} \)
37 \( 1 + 6.60T + 37T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 + 7.34T + 43T^{2} \)
47 \( 1 + 9.34iT - 47T^{2} \)
53 \( 1 + 1.25T + 53T^{2} \)
59 \( 1 + 3.34iT - 59T^{2} \)
61 \( 1 - 7.74iT - 61T^{2} \)
67 \( 1 + 13.3T + 67T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + 13.1iT - 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 - 0.240T + 83T^{2} \)
89 \( 1 - 5.46T + 89T^{2} \)
97 \( 1 + 15.7iT - 97T^{2} \)
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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

−9.201952716647479608979358935552, −8.665787840988010203719112518484, −8.193027063818089925244563354062, −7.14215116549372992515484321059, −5.80994911114585776613428232924, −4.97533949310440952565896708257, −3.60495063430323089381851367715, −3.17500601963073178420832320549, −1.73520152090833724753276676996, −0.07468744492944562247097260328, 1.46427242159013349632034081197, 3.57572259040409502944758257313, 4.37607303529974154533415695311, 5.08149203146562876962727466512, 6.59271615069532305612810420771, 7.06498180001680854102094439069, 7.72325383781508338099008021027, 8.482981891711613835677294493911, 9.401049256482484133456908025185, 10.50306356290074225823333628157

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