Properties

Label 2-1440-5.4-c1-0-6
Degree $2$
Conductor $1440$
Sign $0.447 - 0.894i$
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 − i)5-s + 4i·13-s − 2i·17-s + (3 + 4i)25-s − 4·29-s + 12i·37-s + 8·41-s + 7·49-s + 14i·53-s + 10·61-s + (4 − 8i)65-s + 16i·73-s + (−2 + 4i)85-s + 16·89-s + 8i·97-s + ⋯
L(s)  = 1  + (−0.894 − 0.447i)5-s + 1.10i·13-s − 0.485i·17-s + (0.600 + 0.800i)25-s − 0.742·29-s + 1.97i·37-s + 1.24·41-s + 49-s + 1.92i·53-s + 1.28·61-s + (0.496 − 0.992i)65-s + 1.87i·73-s + (−0.216 + 0.433i)85-s + 1.69·89-s + 0.812i·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.087861922\)
\(L(\frac12)\) \(\approx\) \(1.087861922\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + i)T \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 12iT - 37T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 14iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 16iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 16T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−9.458679988768253084379139758445, −8.925301369306121627365714418917, −8.050081986763397716150557060162, −7.31092479107903217698039894091, −6.54525525749204763048290778765, −5.41502213200377348833706332231, −4.49710116028780179666916488972, −3.83470254322702316279446150214, −2.61159662032030680773899388489, −1.15705028952187128664649683041, 0.50473177986945510754361832342, 2.27971455438168325708404875579, 3.41091975851096070929084056928, 4.08541305420778551649624752280, 5.26633385497965809184663136883, 6.09046397723231810391365551442, 7.14792364725153639479603561366, 7.73887529773438920537607144060, 8.453381622514453753375831803736, 9.357567053701169161926616503547

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