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Properties

Label 2-1440-40.3-c1-0-0
Degree $2$
Conductor $1440$
Sign $-0.987 - 0.156i$
Analytic cond. $11.4984$
Root an. cond. $3.39093$
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  + (1.17 + 1.90i)5-s + (−1.90 − 1.90i)7-s − 3.23·11-s + (−0.726 + 0.726i)13-s + (1 − i)17-s − 2i·19-s + (−4.25 + 4.25i)23-s + (−2.23 + 4.47i)25-s − 6.15·29-s + 8.50i·31-s + (1.38 − 5.85i)35-s + (0.726 + 0.726i)37-s − 5.70·41-s + (−4.61 − 4.61i)43-s + (−3.35 − 3.35i)47-s + ⋯
L(s)  = 1  + (0.525 + 0.850i)5-s + (−0.718 − 0.718i)7-s − 0.975·11-s + (−0.201 + 0.201i)13-s + (0.242 − 0.242i)17-s − 0.458i·19-s + (−0.886 + 0.886i)23-s + (−0.447 + 0.894i)25-s − 1.14·29-s + 1.52i·31-s + (0.233 − 0.989i)35-s + (0.119 + 0.119i)37-s − 0.891·41-s + (−0.704 − 0.704i)43-s + (−0.489 − 0.489i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.987 - 0.156i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (1423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -0.987 - 0.156i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3008738087\)
\(L(\frac12)\) \(\approx\) \(0.3008738087\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-1.17 - 1.90i)T \)
good7 \( 1 + (1.90 + 1.90i)T + 7iT^{2} \)
11 \( 1 + 3.23T + 11T^{2} \)
13 \( 1 + (0.726 - 0.726i)T - 13iT^{2} \)
17 \( 1 + (-1 + i)T - 17iT^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + (4.25 - 4.25i)T - 23iT^{2} \)
29 \( 1 + 6.15T + 29T^{2} \)
31 \( 1 - 8.50iT - 31T^{2} \)
37 \( 1 + (-0.726 - 0.726i)T + 37iT^{2} \)
41 \( 1 + 5.70T + 41T^{2} \)
43 \( 1 + (4.61 + 4.61i)T + 43iT^{2} \)
47 \( 1 + (3.35 + 3.35i)T + 47iT^{2} \)
53 \( 1 + (3.07 - 3.07i)T - 53iT^{2} \)
59 \( 1 - 0.472iT - 59T^{2} \)
61 \( 1 - 0.898iT - 61T^{2} \)
67 \( 1 + (-4.61 + 4.61i)T - 67iT^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + (4.70 + 4.70i)T + 73iT^{2} \)
79 \( 1 - 2.90T + 79T^{2} \)
83 \( 1 + (6.61 + 6.61i)T + 83iT^{2} \)
89 \( 1 - 2.47iT - 89T^{2} \)
97 \( 1 + (-4.23 + 4.23i)T - 97iT^{2} \)
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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

−10.01405687865554196389095349286, −9.356966811864702878606037685820, −8.214723933164858392184943636870, −7.24032360499703722970656986101, −6.86536276235999027491412976037, −5.82195547257718834319728213952, −5.04413973115081589342082193658, −3.69162575018636466148901046039, −3.00514135501531448087805155874, −1.83008863513633611125348834168, 0.10931906589435550009178894541, 1.87730338437352666635707625514, 2.80700530170342081530593735055, 4.05197386986743177831491565557, 5.13466611431671396757255158740, 5.79285842159127521563165105459, 6.44815682329901650408683068459, 7.82682802387477773293777557479, 8.274889962544133652979932788307, 9.304078028809018547913574232580

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