Properties

Label 2-1440-9.4-c1-0-43
Degree $2$
Conductor $1440$
Sign $-0.224 + 0.974i$
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  + (1.69 + 0.344i)3-s + (0.5 − 0.866i)5-s + (−1.69 − 2.93i)7-s + (2.76 + 1.17i)9-s + (−1.84 − 3.19i)11-s + (0.867 − 1.50i)13-s + (1.14 − 1.29i)15-s − 5.93·17-s − 5.10·19-s + (−1.86 − 5.57i)21-s + (2.43 − 4.21i)23-s + (−0.499 − 0.866i)25-s + (4.28 + 2.93i)27-s + (−0.600 − 1.03i)29-s + (2.28 − 3.95i)31-s + ⋯
L(s)  = 1  + (0.979 + 0.199i)3-s + (0.223 − 0.387i)5-s + (−0.641 − 1.11i)7-s + (0.920 + 0.390i)9-s + (−0.556 − 0.963i)11-s + (0.240 − 0.416i)13-s + (0.296 − 0.335i)15-s − 1.43·17-s − 1.17·19-s + (−0.407 − 1.21i)21-s + (0.507 − 0.878i)23-s + (−0.0999 − 0.173i)25-s + (0.824 + 0.565i)27-s + (−0.111 − 0.193i)29-s + (0.410 − 0.710i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.224 + 0.974i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -0.224 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.770120858\)
\(L(\frac12)\) \(\approx\) \(1.770120858\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 - 0.344i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (1.69 + 2.93i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.84 + 3.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.867 + 1.50i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.93T + 17T^{2} \)
19 \( 1 + 5.10T + 19T^{2} \)
23 \( 1 + (-2.43 + 4.21i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.600 + 1.03i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.28 + 3.95i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.14T + 37T^{2} \)
41 \( 1 + (4.32 - 7.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.55 - 6.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.95 + 3.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.25T + 53T^{2} \)
59 \( 1 + (-1.39 + 2.41i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.0729 - 0.126i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.54 + 6.13i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.110T + 71T^{2} \)
73 \( 1 - 15.3T + 73T^{2} \)
79 \( 1 + (2.39 + 4.14i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.38 + 14.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.37T + 89T^{2} \)
97 \( 1 + (-3.76 - 6.51i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.167014738141315568297136303605, −8.517757626718933020369813150744, −7.931708349619539781906531938351, −6.84914541165713366192956349277, −6.22568484223823611739288248887, −4.82695102304877497344424497550, −4.10621593262219146420856635953, −3.19136865538025780828537951360, −2.19790282056933061908172771617, −0.58219474252397568125498841929, 2.00994378752027505111688979161, 2.45899394055500928138928362326, 3.57886706760004319702184470611, 4.60347632313797062083826730884, 5.72735561426732224690428767553, 6.82502395010699608489112545153, 7.09586451268194921541237808742, 8.471349510640990879742705793038, 8.851055577080057895343821871519, 9.601202587945285847905570654650

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