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Properties

Label 2-2160-9.7-c1-0-4
Degree $2$
Conductor $2160$
Sign $-0.649 - 0.760i$
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  + (0.5 + 0.866i)5-s + (0.714 − 1.23i)7-s + (−1.33 + 2.31i)11-s + (−2.33 − 4.04i)13-s − 2.67·17-s − 4.67·19-s + (2.95 + 5.12i)23-s + (−0.499 + 0.866i)25-s + (−4.74 + 8.21i)29-s + (3.48 + 6.02i)31-s + 1.42·35-s − 1.81·37-s + (−0.735 − 1.27i)41-s + (0.235 − 0.408i)43-s + (−3.47 + 6.02i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (0.269 − 0.467i)7-s + (−0.402 + 0.697i)11-s + (−0.648 − 1.12i)13-s − 0.648·17-s − 1.07·19-s + (0.616 + 1.06i)23-s + (−0.0999 + 0.173i)25-s + (−0.881 + 1.52i)29-s + (0.625 + 1.08i)31-s + 0.241·35-s − 0.298·37-s + (−0.114 − 0.198i)41-s + (0.0359 − 0.0622i)43-s + (−0.507 + 0.878i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.649 - 0.760i$
Analytic conductor: \(17.2476\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ -0.649 - 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7925398253\)
\(L(\frac12)\) \(\approx\) \(0.7925398253\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (-0.714 + 1.23i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.33 - 2.31i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.33 + 4.04i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.67T + 17T^{2} \)
19 \( 1 + 4.67T + 19T^{2} \)
23 \( 1 + (-2.95 - 5.12i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.74 - 8.21i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.48 - 6.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.81T + 37T^{2} \)
41 \( 1 + (0.735 + 1.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.235 + 0.408i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.47 - 6.02i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.14T + 53T^{2} \)
59 \( 1 + (-0.571 - 0.990i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.26 + 2.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.29 + 5.70i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 + 1.71T + 73T^{2} \)
79 \( 1 + (0.143 - 0.249i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.14 + 3.71i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (3.91 - 6.78i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.351894658273116638637629270360, −8.609209504431636996646934241533, −7.58099989762063630681816689610, −7.21416992886951013493417121297, −6.28976277107834663326132543802, −5.24752701704639855668487123003, −4.68280171485324491125478736029, −3.50804649588930126005216584034, −2.61064903605417637997800807901, −1.49654772680236518713677970147, 0.25933863397090701527384097998, 1.96975095058525309127972335769, 2.60653625301614893075283138271, 4.10589337428780315402878283856, 4.67117100383566117267687263040, 5.67221664882227364919256247531, 6.37221592980921217494497287171, 7.19686145955159001371054819107, 8.291797839897935872105012270550, 8.681182728184626153746832864630

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