Properties

Label 2-1440-15.14-c0-0-3
Degree $2$
Conductor $1440$
Sign $-0.169 + 0.985i$
Analytic cond. $0.718653$
Root an. cond. $0.847734$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)5-s − 2i·13-s − 1.41·17-s + 1.00i·25-s − 1.41i·29-s − 1.41i·41-s + 49-s + 1.41·53-s − 2·61-s + (−1.41 + 1.41i)65-s − 2i·73-s + (1.00 + 1.00i)85-s + 1.41i·89-s + 2i·97-s + 1.41i·101-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)5-s − 2i·13-s − 1.41·17-s + 1.00i·25-s − 1.41i·29-s − 1.41i·41-s + 49-s + 1.41·53-s − 2·61-s + (−1.41 + 1.41i)65-s − 2i·73-s + (1.00 + 1.00i)85-s + 1.41i·89-s + 2i·97-s + 1.41i·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.169 + 0.985i$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :0),\ -0.169 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7674301355\)
\(L(\frac12)\) \(\approx\) \(0.7674301355\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 2iT - T^{2} \)
17 \( 1 + 1.41T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 2T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 2iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 - 2iT - T^{2} \)
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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

−9.348997356311096425505065108385, −8.641718453927909592263013396562, −7.932405047998646393353733847485, −7.30476391923373773354794643674, −6.10737132164266983510691258898, −5.30183951511876939799916108282, −4.41983372176674333045357554816, −3.52914921997263042770272775758, −2.35102325481535181550146216901, −0.62134370471729405049938759768, 1.84536657162617498568943779418, 2.96049967869702040466268664607, 4.12683028261062857923536465982, 4.61925390597054316659859111273, 6.06310372854863886046778034149, 6.91140905814811315556539473835, 7.22874385489516996086569058046, 8.515600871157067927999478252506, 8.994557651964444412992615570431, 9.953044184929803010568313884903

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