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Properties

Label 2-1080-40.29-c1-0-84
Degree $2$
Conductor $1080$
Sign $0.354 + 0.935i$
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  + 1.41·2-s + 2.00·4-s + (−0.792 − 2.09i)5-s − 0.717i·7-s + 2.82·8-s + (−1.12 − 2.95i)10-s − 3.16i·11-s − 1.01i·14-s + 4.00·16-s + (−1.58 − 4.18i)20-s − 4.47i·22-s + (−3.74 + 3.31i)25-s − 1.43i·28-s − 10.3i·29-s − 0.757·31-s + 5.65·32-s + ⋯
L(s)  = 1  + 1.00·2-s + 1.00·4-s + (−0.354 − 0.935i)5-s − 0.271i·7-s + 1.00·8-s + (−0.354 − 0.935i)10-s − 0.954i·11-s − 0.271i·14-s + 1.00·16-s + (−0.354 − 0.935i)20-s − 0.954i·22-s + (−0.748 + 0.663i)25-s − 0.271i·28-s − 1.92i·29-s − 0.136·31-s + 1.00·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.354 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.354 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.354 + 0.935i$
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: \((2,\ 1080,\ (\ :1/2),\ 0.354 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.867610356\)
\(L(\frac12)\) \(\approx\) \(2.867610356\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 \)
5 \( 1 + (0.792 + 2.09i)T \)
good7 \( 1 + 0.717iT - 7T^{2} \)
11 \( 1 + 3.16iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 10.3iT - 29T^{2} \)
31 \( 1 + 0.757T + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 17.0iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 8.06iT - 97T^{2} \)
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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

−9.833121831898278286328918483504, −8.726968791425795642147645701084, −7.987982895067567638716497438043, −7.16535593264321197275013763858, −6.03049482267964768227148087819, −5.42711927250927920450498141208, −4.34957476433615146161598515773, −3.74574349041979696193622002586, −2.46633477721850670044807430316, −0.954180474857727631308630951056, 1.88698072817727331270433788293, 2.92118570596860189924075230221, 3.81121222313899292083872701241, 4.79667600347633090632351756607, 5.71389288753232826704802898649, 6.76709491719538137023362591994, 7.18928437221459298800737633874, 8.110590517377587721427845134980, 9.334869516732419075018889926355, 10.40978073433818793931945511656

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