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Properties

Label 2-2160-240.149-c0-0-5
Degree $2$
Conductor $2160$
Sign $0.991 - 0.130i$
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  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 − 0.707i)5-s + (0.707 + 0.707i)8-s + (−0.500 − 0.866i)10-s + (0.500 + 0.866i)16-s + 0.517·17-s + (1.36 − 1.36i)19-s + (−0.258 − 0.965i)20-s + 0.517i·23-s + 1.00i·25-s + 1.73·31-s + (0.258 + 0.965i)32-s + (0.499 + 0.133i)34-s + (1.67 − 0.965i)38-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 − 0.707i)5-s + (0.707 + 0.707i)8-s + (−0.500 − 0.866i)10-s + (0.500 + 0.866i)16-s + 0.517·17-s + (1.36 − 1.36i)19-s + (−0.258 − 0.965i)20-s + 0.517i·23-s + 1.00i·25-s + 1.73·31-s + (0.258 + 0.965i)32-s + (0.499 + 0.133i)34-s + (1.67 − 0.965i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.991 - 0.130i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :0),\ 0.991 - 0.130i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.020587557\)
\(L(\frac12)\) \(\approx\) \(2.020587557\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + T^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - 0.517T + T^{2} \)
19 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
23 \( 1 - 0.517iT - T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 - 1.73T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
89 \( 1 + T^{2} \)
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

−9.187682203148868289193673424541, −8.228315189664579145157785561193, −7.67960707804165621390199472094, −6.93434460139263097614863614342, −6.03400900501140486274073434248, −4.98687878794457307218893570574, −4.71770579692872513338808996835, −3.54067361044195179921401347509, −2.88293783351315656184906670919, −1.32986421537821225354457699234, 1.43077858161273097815457052874, 2.85787584510967764741605991585, 3.37765107572323775135008105252, 4.31249022713211074496259772437, 5.14136201317314403099554303208, 6.15320985926126871135944656160, 6.67599528886480564516435730746, 7.77141944380481532382919997268, 8.015506872958612472310954824371, 9.570154790350715105550016537471

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