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Properties

Label 2-2160-180.139-c0-0-1
Degree $2$
Conductor $2160$
Sign $0.173 + 0.984i$
Analytic cond. $1.07798$
Root an. cond. $1.03825$
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  + (−0.5 + 0.866i)5-s + (−0.866 − 1.5i)7-s + (0.866 − 1.5i)23-s + (−0.499 − 0.866i)25-s + (−0.5 − 0.866i)29-s + 1.73·35-s + (0.5 − 0.866i)41-s + (−0.866 − 1.5i)47-s + (−1 + 1.73i)49-s + (−0.5 − 0.866i)61-s + (0.866 − 1.5i)67-s + (0.866 + 1.5i)83-s + 89-s + (1 + 1.73i)101-s − 1.73·107-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s + (−0.866 − 1.5i)7-s + (0.866 − 1.5i)23-s + (−0.499 − 0.866i)25-s + (−0.5 − 0.866i)29-s + 1.73·35-s + (0.5 − 0.866i)41-s + (−0.866 − 1.5i)47-s + (−1 + 1.73i)49-s + (−0.5 − 0.866i)61-s + (0.866 − 1.5i)67-s + (0.866 + 1.5i)83-s + 89-s + (1 + 1.73i)101-s − 1.73·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :0),\ 0.173 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7671400697\)
\(L(\frac12)\) \(\approx\) \(0.7671400697\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.207830977810224538605690992059, −8.103574727267180104440215752817, −7.46621794624362066218790221929, −6.70061787975834571015686303652, −6.36389806479051389419857020897, −4.94316220882254008543057175965, −3.95832788026268393827972234958, −3.45474718625286887030640649981, −2.38439582429377656778591876945, −0.55141582021464210397343619677, 1.48127986216760581996466882832, 2.81541551353008979559568852167, 3.57165095179946575183583520492, 4.76348182084597636064235446332, 5.48617037245777466407456636448, 6.12640977067959197527110102804, 7.17822070982291029772499092537, 7.985956267914304075923238843884, 8.880721768977359384084929873906, 9.224665982574110897268308440571

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