Properties

Label 2-2880-320.19-c0-0-1
Degree $2$
Conductor $2880$
Sign $-0.290 + 0.956i$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
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  + (0.195 − 0.980i)2-s + (−0.923 − 0.382i)4-s + (0.980 − 0.195i)5-s + (−0.555 + 0.831i)8-s i·10-s + (0.707 + 0.707i)16-s + (0.275 − 0.275i)17-s + (0.216 − 1.08i)19-s + (−0.980 − 0.195i)20-s + (−0.360 − 0.149i)23-s + (0.923 − 0.382i)25-s + 0.765·31-s + (0.831 − 0.555i)32-s + (−0.216 − 0.324i)34-s + (−1.02 − 0.425i)38-s + ⋯
L(s)  = 1  + (0.195 − 0.980i)2-s + (−0.923 − 0.382i)4-s + (0.980 − 0.195i)5-s + (−0.555 + 0.831i)8-s i·10-s + (0.707 + 0.707i)16-s + (0.275 − 0.275i)17-s + (0.216 − 1.08i)19-s + (−0.980 − 0.195i)20-s + (−0.360 − 0.149i)23-s + (0.923 − 0.382i)25-s + 0.765·31-s + (0.831 − 0.555i)32-s + (−0.216 − 0.324i)34-s + (−1.02 − 0.425i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.290 + 0.956i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ -0.290 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.449632723\)
\(L(\frac12)\) \(\approx\) \(1.449632723\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.195 + 0.980i)T \)
3 \( 1 \)
5 \( 1 + (-0.980 + 0.195i)T \)
good7 \( 1 + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.382 - 0.923i)T^{2} \)
13 \( 1 + (0.923 + 0.382i)T^{2} \)
17 \( 1 + (-0.275 + 0.275i)T - iT^{2} \)
19 \( 1 + (-0.216 + 1.08i)T + (-0.923 - 0.382i)T^{2} \)
23 \( 1 + (0.360 + 0.149i)T + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.382 - 0.923i)T^{2} \)
31 \( 1 - 0.765T + T^{2} \)
37 \( 1 + (-0.923 + 0.382i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (-0.382 - 0.923i)T^{2} \)
47 \( 1 + (1.17 + 1.17i)T + iT^{2} \)
53 \( 1 + (-0.425 + 0.636i)T + (-0.382 - 0.923i)T^{2} \)
59 \( 1 + (0.923 - 0.382i)T^{2} \)
61 \( 1 + (-0.923 - 1.38i)T + (-0.382 + 0.923i)T^{2} \)
67 \( 1 + (-0.382 + 0.923i)T^{2} \)
71 \( 1 + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (-1 - i)T + iT^{2} \)
83 \( 1 + (1.81 + 0.360i)T + (0.923 + 0.382i)T^{2} \)
89 \( 1 + (0.707 - 0.707i)T^{2} \)
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

−8.833566475842304817866897666987, −8.386720729618673554761170061905, −7.13542342898798721357409164704, −6.28352054518500943376987264412, −5.38967481876352892454143250051, −4.88863668361951704718900342249, −3.88752705388226952062975612285, −2.82817851513035016507267637672, −2.13472417180944745015812404986, −0.970293690920145369487586899141, 1.43410570224956498401158847574, 2.79344821455558948114929875385, 3.75778753487510310125775440283, 4.71427766147394240544939091377, 5.57344680239494113374215131791, 6.09878836324473637457865486522, 6.75614343056475302995062757470, 7.67519834681296054237947888171, 8.269061982162761179117217909543, 9.120925054211771796423963102063

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