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Properties

Label 2-1440-120.59-c1-0-21
Degree $2$
Conductor $1440$
Sign $-0.577 + 0.816i$
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  − 2.23i·5-s − 1.16·7-s − 5.88i·11-s + 7.16·13-s − 6.32·19-s + 4.47i·23-s − 5.00·25-s + 2.59i·35-s − 4.83·37-s − 7.53i·41-s − 2.82i·47-s − 5.64·49-s − 5.65i·53-s − 13.1·55-s − 14.3i·59-s + ⋯
L(s)  = 1  − 0.999i·5-s − 0.439·7-s − 1.77i·11-s + 1.98·13-s − 1.45·19-s + 0.932i·23-s − 1.00·25-s + 0.439i·35-s − 0.795·37-s − 1.17i·41-s − 0.412i·47-s − 0.807·49-s − 0.777i·53-s − 1.77·55-s − 1.87i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.266943369\)
\(L(\frac12)\) \(\approx\) \(1.266943369\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23iT \)
good7 \( 1 + 1.16T + 7T^{2} \)
11 \( 1 + 5.88iT - 11T^{2} \)
13 \( 1 - 7.16T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 6.32T + 19T^{2} \)
23 \( 1 - 4.47iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 4.83T + 37T^{2} \)
41 \( 1 + 7.53iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 2.82iT - 47T^{2} \)
53 \( 1 + 5.65iT - 53T^{2} \)
59 \( 1 + 14.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 0.955iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−8.823882719917764146031263215948, −8.710923294568190289269788426018, −7.957446300353468450842693887626, −6.51778234158045449930650627730, −5.98977912848124216237595575197, −5.21815837402600005450359929857, −3.87326946974463255462457844117, −3.42568832698063406872758311179, −1.71641290487356399245716768086, −0.51085626317890080471003264007, 1.69120003800338443583909468310, 2.75283549626906864578815276818, 3.84605364044231014036184512600, 4.56403163020466382458687326896, 6.04643362070332428347895064793, 6.50190444980490579586768398182, 7.23417919187770277813885074598, 8.217714219273337131041999049350, 9.009299602724291616700693338763, 9.978658389324202082360402687552

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