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Properties

Label 2-2160-45.34-c1-0-24
Degree $2$
Conductor $2160$
Sign $0.874 + 0.484i$
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  + (1.30 + 1.81i)5-s + (0.389 + 0.224i)7-s + (1.72 − 2.98i)11-s + (2.12 − 1.22i)13-s − 5.89i·17-s − 5.44·19-s + (5.97 − 3.44i)23-s + (−1.58 + 4.74i)25-s + (3 − 5.19i)29-s + (0.775 + 1.34i)31-s + (0.101 + i)35-s − 8i·37-s + (−0.5 − 0.866i)41-s + (2.20 + 1.27i)43-s + (−3.85 − 2.22i)47-s + ⋯
L(s)  = 1  + (0.584 + 0.811i)5-s + (0.147 + 0.0849i)7-s + (0.520 − 0.900i)11-s + (0.588 − 0.339i)13-s − 1.43i·17-s − 1.25·19-s + (1.24 − 0.719i)23-s + (−0.316 + 0.948i)25-s + (0.557 − 0.964i)29-s + (0.139 + 0.241i)31-s + (0.0170 + 0.169i)35-s − 1.31i·37-s + (−0.0780 − 0.135i)41-s + (0.336 + 0.194i)43-s + (−0.562 − 0.324i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.874 + 0.484i$
Analytic conductor: \(17.2476\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ 0.874 + 0.484i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.054409306\)
\(L(\frac12)\) \(\approx\) \(2.054409306\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 - 1.81i)T \)
good7 \( 1 + (-0.389 - 0.224i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.72 + 2.98i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.12 + 1.22i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.89iT - 17T^{2} \)
19 \( 1 + 5.44T + 19T^{2} \)
23 \( 1 + (-5.97 + 3.44i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.775 - 1.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.20 - 1.27i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.85 + 2.22i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.55iT - 53T^{2} \)
59 \( 1 + (-6.62 - 11.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.22 - 3.85i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.94 + 2.27i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.44T + 71T^{2} \)
73 \( 1 - 14.7iT - 73T^{2} \)
79 \( 1 + (3.67 - 6.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.46 - 2i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.10T + 89T^{2} \)
97 \( 1 + (-11.2 - 6.5i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.908336851983551386662843157962, −8.440265166821725958105990012610, −7.28760964071675369059462231041, −6.63392545573513410558756704341, −5.97225343429799253089276843345, −5.13713289575320308290035158181, −4.02677159397045226010376381335, −3.03460031398382294750372517169, −2.28580424827526547241870255093, −0.789900959862413405976091131625, 1.30227084618219156570852623925, 1.95569389425472921826368921083, 3.43390357182837179683857963683, 4.45576491728086418009454173919, 4.96490362888112900705023222714, 6.18455546973936327929200752675, 6.54229183034826435834868779921, 7.69126869235847530279761338215, 8.577257358656570768371293529909, 8.994163614256173038711420416497

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