Properties

Label 2-1440-120.53-c1-0-10
Degree $2$
Conductor $1440$
Sign $0.998 + 0.0621i$
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  + (1.42 − 1.72i)5-s + (−3.11 + 3.11i)7-s + 1.17·11-s + (2.15 − 2.15i)13-s + (1.33 + 1.33i)17-s − 0.322·19-s + (4.71 − 4.71i)23-s + (−0.933 − 4.91i)25-s + 6.63i·29-s + 0.0675·31-s + (0.924 + 9.82i)35-s + (7.60 + 7.60i)37-s − 3.19i·41-s + (6.70 − 6.70i)43-s + (7.34 + 7.34i)47-s + ⋯
L(s)  = 1  + (0.637 − 0.770i)5-s + (−1.17 + 1.17i)7-s + 0.355·11-s + (0.596 − 0.596i)13-s + (0.323 + 0.323i)17-s − 0.0739·19-s + (0.982 − 0.982i)23-s + (−0.186 − 0.982i)25-s + 1.23i·29-s + 0.0121·31-s + (0.156 + 1.66i)35-s + (1.25 + 1.25i)37-s − 0.499i·41-s + (1.02 − 1.02i)43-s + (1.07 + 1.07i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.998 + 0.0621i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ 0.998 + 0.0621i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.755839274\)
\(L(\frac12)\) \(\approx\) \(1.755839274\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.42 + 1.72i)T \)
good7 \( 1 + (3.11 - 3.11i)T - 7iT^{2} \)
11 \( 1 - 1.17T + 11T^{2} \)
13 \( 1 + (-2.15 + 2.15i)T - 13iT^{2} \)
17 \( 1 + (-1.33 - 1.33i)T + 17iT^{2} \)
19 \( 1 + 0.322T + 19T^{2} \)
23 \( 1 + (-4.71 + 4.71i)T - 23iT^{2} \)
29 \( 1 - 6.63iT - 29T^{2} \)
31 \( 1 - 0.0675T + 31T^{2} \)
37 \( 1 + (-7.60 - 7.60i)T + 37iT^{2} \)
41 \( 1 + 3.19iT - 41T^{2} \)
43 \( 1 + (-6.70 + 6.70i)T - 43iT^{2} \)
47 \( 1 + (-7.34 - 7.34i)T + 47iT^{2} \)
53 \( 1 + (-5.73 - 5.73i)T + 53iT^{2} \)
59 \( 1 - 8.68iT - 59T^{2} \)
61 \( 1 + 12.5iT - 61T^{2} \)
67 \( 1 + (1.87 + 1.87i)T + 67iT^{2} \)
71 \( 1 + 4.18iT - 71T^{2} \)
73 \( 1 + (-3.97 - 3.97i)T + 73iT^{2} \)
79 \( 1 + 9.66iT - 79T^{2} \)
83 \( 1 + (-0.585 - 0.585i)T + 83iT^{2} \)
89 \( 1 + 0.557T + 89T^{2} \)
97 \( 1 + (10.5 - 10.5i)T - 97iT^{2} \)
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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

−9.233039749650334159075722669893, −8.983489368259996651092089341697, −8.183185222140439597774777585634, −6.88744338692725476365197468468, −6.03952524315196006565901901528, −5.61784159788549163371253875133, −4.55157767756015231541832753625, −3.28864964029817115292031674394, −2.43331214448077604017805426961, −0.988112987464459672054073386795, 0.979414146537511358967198134953, 2.49766263141304457747385017734, 3.52438678260817968976871200715, 4.18102620764510571446512427706, 5.65732474578972691052078032114, 6.35892810848224907947766924288, 7.04900413407542347076539469699, 7.63084140525859667776837490528, 9.043280013846904849937944810719, 9.634460725530980012670229155314

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