Properties

Label 2-1440-5.4-c1-0-9
Degree $2$
Conductor $1440$
Sign $0.894 - 0.447i$
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 − 2i)5-s + 4i·13-s + 8i·17-s + (−3 + 4i)25-s + 10·29-s − 12i·37-s + 10·41-s + 7·49-s + 4i·53-s + 10·61-s + (8 − 4i)65-s + 16i·73-s + (16 − 8i)85-s − 10·89-s + 8i·97-s + ⋯
L(s)  = 1  + (−0.447 − 0.894i)5-s + 1.10i·13-s + 1.94i·17-s + (−0.600 + 0.800i)25-s + 1.85·29-s − 1.97i·37-s + 1.56·41-s + 49-s + 0.549i·53-s + 1.28·61-s + (0.992 − 0.496i)65-s + 1.87i·73-s + (1.73 − 0.867i)85-s − 1.05·89-s + 0.812i·97-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.894 - 0.447i$
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: \((2,\ 1440,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.431706806\)
\(L(\frac12)\) \(\approx\) \(1.431706806\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 2i)T \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 8iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 12iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 4iT - 53T^{2} \)
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 \( 1 + 10T + 89T^{2} \)
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

−9.420844271631511952858485504146, −8.724196582595219545465983625642, −8.162169145789519053179197596571, −7.22451300132872997519292084108, −6.26634789343380574013748111604, −5.45167070235756735737882026857, −4.27324654952734499041190090775, −3.93107504365283413400261443332, −2.30702506626782030106039441031, −1.10903140412871219835651823945, 0.70081637769018333116178111448, 2.65571018849977082810313264801, 3.11797907156419186966667701453, 4.40214920062796692947090991854, 5.27114339318429747720415150274, 6.34110085891932730965192004051, 7.07682236961043110027976443929, 7.79941900119667763682971898495, 8.534964065312342776141551775522, 9.654155887792734734389719725953

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