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Properties

Label 2-2160-15.8-c1-0-24
Degree $2$
Conductor $2160$
Sign $0.940 + 0.340i$
Analytic cond. $17.2476$
Root an. cond. $4.15303$
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  + (−1.87 − 1.22i)5-s + (1.79 − 1.79i)7-s + 0.646i·11-s + (3.79 + 3.79i)13-s + (4.96 + 4.96i)17-s − 6.58i·19-s + (−1.87 + 1.87i)23-s + (2 + 4.58i)25-s − 7.99·29-s + 0.582·31-s + (−5.54 + 1.15i)35-s + (3 − 3i)37-s + 4.25i·41-s + (0.791 + 0.791i)43-s + (1.93 + 1.93i)47-s + ⋯
L(s)  = 1  + (−0.836 − 0.547i)5-s + (0.677 − 0.677i)7-s + 0.194i·11-s + (1.05 + 1.05i)13-s + (1.20 + 1.20i)17-s − 1.51i·19-s + (−0.390 + 0.390i)23-s + (0.400 + 0.916i)25-s − 1.48·29-s + 0.104·31-s + (−0.937 + 0.195i)35-s + (0.493 − 0.493i)37-s + 0.664i·41-s + (0.120 + 0.120i)43-s + (0.282 + 0.282i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.940 + 0.340i$
Analytic conductor: \(17.2476\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ 0.940 + 0.340i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.736400316\)
\(L(\frac12)\) \(\approx\) \(1.736400316\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.87 + 1.22i)T \)
good7 \( 1 + (-1.79 + 1.79i)T - 7iT^{2} \)
11 \( 1 - 0.646iT - 11T^{2} \)
13 \( 1 + (-3.79 - 3.79i)T + 13iT^{2} \)
17 \( 1 + (-4.96 - 4.96i)T + 17iT^{2} \)
19 \( 1 + 6.58iT - 19T^{2} \)
23 \( 1 + (1.87 - 1.87i)T - 23iT^{2} \)
29 \( 1 + 7.99T + 29T^{2} \)
31 \( 1 - 0.582T + 31T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 - 4.25iT - 41T^{2} \)
43 \( 1 + (-0.791 - 0.791i)T + 43iT^{2} \)
47 \( 1 + (-1.93 - 1.93i)T + 47iT^{2} \)
53 \( 1 + (-5.47 + 5.47i)T - 53iT^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 - 6.16T + 61T^{2} \)
67 \( 1 + (-7 + 7i)T - 67iT^{2} \)
71 \( 1 + 14.3iT - 71T^{2} \)
73 \( 1 + (1.20 + 1.20i)T + 73iT^{2} \)
79 \( 1 - 6.16iT - 79T^{2} \)
83 \( 1 + (-1.22 + 1.22i)T - 83iT^{2} \)
89 \( 1 - 9.42T + 89T^{2} \)
97 \( 1 + (-7.58 + 7.58i)T - 97iT^{2} \)
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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

−8.939615383236145179616475476288, −8.187239889697910092429720864024, −7.60692429572474737700590528842, −6.85181675133995603736631234487, −5.82123679691109863087096408840, −4.88049589291396851096068973052, −4.05465138800125673199708492373, −3.57514532136000761063903429909, −1.87989570709029674959854913832, −0.893066430654483616123088564872, 0.914919136683459442777036337991, 2.37141182793316391546035290000, 3.40997239420174553189632363347, 4.00409636164300009720019469871, 5.43905393668035780762782475702, 5.68682956929277860318191508623, 6.89617041537010784718570953679, 7.83329900265970622586491961774, 8.116984105094759980597908349100, 8.933117615460227697322585439392

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