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Properties

Label 2-1080-40.29-c1-0-34
Degree $2$
Conductor $1080$
Sign $0.999 + 0.0211i$
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  + (−1.35 − 0.399i)2-s + (1.68 + 1.08i)4-s + (−1.49 + 1.66i)5-s − 1.43i·7-s + (−1.84 − 2.14i)8-s + (2.69 − 1.65i)10-s + 0.0984i·11-s + 1.92·13-s + (−0.573 + 1.94i)14-s + (1.65 + 3.64i)16-s − 2.03i·17-s − 0.999i·19-s + (−4.31 + 1.17i)20-s + (0.0393 − 0.133i)22-s + 4.30i·23-s + ⋯
L(s)  = 1  + (−0.959 − 0.282i)2-s + (0.840 + 0.541i)4-s + (−0.668 + 0.743i)5-s − 0.542i·7-s + (−0.653 − 0.757i)8-s + (0.851 − 0.524i)10-s + 0.0296i·11-s + 0.533·13-s + (−0.153 + 0.520i)14-s + (0.412 + 0.910i)16-s − 0.494i·17-s − 0.229i·19-s + (−0.965 + 0.262i)20-s + (0.00838 − 0.0284i)22-s + 0.897i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.999 + 0.0211i$
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: \((2,\ 1080,\ (\ :1/2),\ 0.999 + 0.0211i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8860321541\)
\(L(\frac12)\) \(\approx\) \(0.8860321541\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 + 0.399i)T \)
3 \( 1 \)
5 \( 1 + (1.49 - 1.66i)T \)
good7 \( 1 + 1.43iT - 7T^{2} \)
11 \( 1 - 0.0984iT - 11T^{2} \)
13 \( 1 - 1.92T + 13T^{2} \)
17 \( 1 + 2.03iT - 17T^{2} \)
19 \( 1 + 0.999iT - 19T^{2} \)
23 \( 1 - 4.30iT - 23T^{2} \)
29 \( 1 + 3.56iT - 29T^{2} \)
31 \( 1 - 1.42T + 31T^{2} \)
37 \( 1 - 8.27T + 37T^{2} \)
41 \( 1 - 5.11T + 41T^{2} \)
43 \( 1 + 5.50T + 43T^{2} \)
47 \( 1 - 9.00iT - 47T^{2} \)
53 \( 1 - 4.93T + 53T^{2} \)
59 \( 1 + 6.27iT - 59T^{2} \)
61 \( 1 - 12.8iT - 61T^{2} \)
67 \( 1 - 1.89T + 67T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 - 5.62iT - 73T^{2} \)
79 \( 1 - 2.48T + 79T^{2} \)
83 \( 1 - 2.26T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + 14.7iT - 97T^{2} \)
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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

−9.914882688431161228707801608654, −9.146421321580601760322622181230, −8.097941477360394761786841768889, −7.55578531828539094118672593668, −6.81727117693726659272807052306, −5.94915245051040179743279934711, −4.31023792650936276542264508842, −3.43275781417729964041872161230, −2.44826358691542623986815436772, −0.846520807568087908858420314146, 0.821801612302972154974882182272, 2.17478838518341000565804909797, 3.56967605630348859898919439240, 4.82740520838544622742994162049, 5.78468749316477312380411842426, 6.61571247090220265394186777408, 7.65949900016491936340824546415, 8.384604242770033204315878245934, 8.844428867773285139384668215208, 9.687080809813407341520201350581

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