Properties

Label 2-1440-20.7-c1-0-8
Degree $2$
Conductor $1440$
Sign $0.525 - 0.850i$
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 + (−2 + 2i)7-s + (−1 + i)13-s + (5 + 5i)17-s − 4·19-s + (2 + 2i)23-s + (3 − 4i)25-s + 4i·29-s + 4i·31-s + (−2 + 6i)35-s + (1 + i)37-s + (6 + 6i)43-s + (−2 + 2i)47-s i·49-s + (7 − 7i)53-s + ⋯
L(s)  = 1  + (0.894 − 0.447i)5-s + (−0.755 + 0.755i)7-s + (−0.277 + 0.277i)13-s + (1.21 + 1.21i)17-s − 0.917·19-s + (0.417 + 0.417i)23-s + (0.600 − 0.800i)25-s + 0.742i·29-s + 0.718i·31-s + (−0.338 + 1.01i)35-s + (0.164 + 0.164i)37-s + (0.914 + 0.914i)43-s + (−0.291 + 0.291i)47-s − 0.142i·49-s + (0.961 − 0.961i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ 0.525 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.665042471\)
\(L(\frac12)\) \(\approx\) \(1.665042471\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 2i)T - 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (1 - i)T - 13iT^{2} \)
17 \( 1 + (-5 - 5i)T + 17iT^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-2 - 2i)T + 23iT^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + (-1 - i)T + 37iT^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + (-6 - 6i)T + 43iT^{2} \)
47 \( 1 + (2 - 2i)T - 47iT^{2} \)
53 \( 1 + (-7 + 7i)T - 53iT^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + (-10 + 10i)T - 67iT^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + (3 - 3i)T - 73iT^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + (2 + 2i)T + 83iT^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (3 + 3i)T + 97iT^{2} \)
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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

−9.706368679627671892363660205536, −8.866677094136685555256486168361, −8.323829790342609236130197318107, −7.10765299498488898416790245043, −6.18171859291098007877303627515, −5.69542037034675962220846327165, −4.75726961685288684636032462456, −3.52891379735976921078187113415, −2.49104297607341325453529706241, −1.38383062808974293538513952391, 0.70139675791265232929629813906, 2.32131597177425001208347563223, 3.16919141435582834602372661986, 4.24584519399071809216271442701, 5.39838589141116257519069002845, 6.12225147271920256764736816104, 7.01751896806842307494192525828, 7.53825700585659573809609403726, 8.746030901339129844528319207681, 9.634226765099376950973003763797

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