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Properties

Label 2-1440-12.11-c1-0-3
Degree $2$
Conductor $1440$
Sign $0.169 - 0.985i$
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  + i·5-s − 3.41i·7-s − 1.41·11-s − 6.24·13-s + 6.82i·17-s + 5.65i·19-s + 8.82·23-s − 25-s + 2i·29-s + 8.82i·31-s + 3.41·35-s + 7.41·37-s − 0.242i·41-s − 1.17i·43-s + 8·47-s + ⋯
L(s)  = 1  + 0.447i·5-s − 1.29i·7-s − 0.426·11-s − 1.73·13-s + 1.65i·17-s + 1.29i·19-s + 1.84·23-s − 0.200·25-s + 0.371i·29-s + 1.58i·31-s + 0.577·35-s + 1.21·37-s − 0.0378i·41-s − 0.178i·43-s + 1.16·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.169 - 0.985i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ 0.169 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.157647348\)
\(L(\frac12)\) \(\approx\) \(1.157647348\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good7 \( 1 + 3.41iT - 7T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 + 6.24T + 13T^{2} \)
17 \( 1 - 6.82iT - 17T^{2} \)
19 \( 1 - 5.65iT - 19T^{2} \)
23 \( 1 - 8.82T + 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 - 8.82iT - 31T^{2} \)
37 \( 1 - 7.41T + 37T^{2} \)
41 \( 1 + 0.242iT - 41T^{2} \)
43 \( 1 + 1.17iT - 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 1.17iT - 53T^{2} \)
59 \( 1 - 2.58T + 59T^{2} \)
61 \( 1 + 8.82T + 61T^{2} \)
67 \( 1 - 11.3iT - 67T^{2} \)
71 \( 1 - 2.34T + 71T^{2} \)
73 \( 1 + 4.82T + 73T^{2} \)
79 \( 1 + 10.4iT - 79T^{2} \)
83 \( 1 + 6.82T + 83T^{2} \)
89 \( 1 - 15.0iT - 89T^{2} \)
97 \( 1 + 6.48T + 97T^{2} \)
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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

−9.980765724273275077382247536286, −8.915301691997656513763743859491, −7.87515447966018761432954967929, −7.32521522497663049766174179144, −6.65113194989042582094893117062, −5.53994423353073878472801338714, −4.59221170671844044585940677846, −3.71283524961155400856518394413, −2.70262812493251818765259519829, −1.31164535895950684022862687168, 0.48389747022809683399596667945, 2.51623657228594834221155548850, 2.72406702414689232014636958894, 4.67389202536973141680230153908, 5.01147259059228948189460436490, 5.88841709561671701736296299263, 7.12020353405506879437636033365, 7.61074968265372701348720539209, 8.779159043730387808808556640723, 9.361869272222781073832451629947

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