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Properties

Label 2-1440-160.19-c0-0-0
Degree $2$
Conductor $1440$
Sign $0.555 + 0.831i$
Analytic cond. $0.718653$
Root an. cond. $0.847734$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)4-s + (0.382 + 0.923i)5-s + (−0.923 + 0.382i)8-s + 10-s + i·16-s + 1.84·17-s + (0.707 − 1.70i)19-s + (0.382 − 0.923i)20-s + (0.541 + 0.541i)23-s + (−0.707 + 0.707i)25-s − 1.41i·31-s + (0.923 + 0.382i)32-s + (0.707 − 1.70i)34-s + (−1.30 − 1.30i)38-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)4-s + (0.382 + 0.923i)5-s + (−0.923 + 0.382i)8-s + 10-s + i·16-s + 1.84·17-s + (0.707 − 1.70i)19-s + (0.382 − 0.923i)20-s + (0.541 + 0.541i)23-s + (−0.707 + 0.707i)25-s − 1.41i·31-s + (0.923 + 0.382i)32-s + (0.707 − 1.70i)34-s + (−1.30 − 1.30i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.555 + 0.831i$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :0),\ 0.555 + 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.346837721\)
\(L(\frac12)\) \(\approx\) \(1.346837721\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 + 0.923i)T \)
3 \( 1 \)
5 \( 1 + (-0.382 - 0.923i)T \)
good7 \( 1 - iT^{2} \)
11 \( 1 + (0.707 - 0.707i)T^{2} \)
13 \( 1 + (-0.707 - 0.707i)T^{2} \)
17 \( 1 - 1.84T + T^{2} \)
19 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
23 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
29 \( 1 + (-0.707 - 0.707i)T^{2} \)
31 \( 1 + 1.41iT - T^{2} \)
37 \( 1 + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (-0.707 + 0.707i)T^{2} \)
47 \( 1 - 1.84iT - T^{2} \)
53 \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 + 0.707i)T^{2} \)
61 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 + (-0.707 - 0.707i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 2T + T^{2} \)
83 \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 - T^{2} \)
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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

−9.531555225639342006175746437668, −9.399904678785301262628204516463, −7.950480890242577609284118807644, −7.17070348572596042979740311329, −6.06336744428075710689591055225, −5.43551914263728054837150957472, −4.40616895923143626590892893629, −3.19223574811498754235292801754, −2.75411192494407367358931728483, −1.32239118898914464723366804000, 1.35326322453470757397759246155, 3.17184652890364860468629258192, 4.01966055549181989290746870474, 5.29051211760914600236031270383, 5.43445196039847682003895836933, 6.50378277082367548913192166125, 7.48493252972116377867043804991, 8.219626761502157695984613077385, 8.773609955791345383231418163240, 9.822545425116588180613369880299

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