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Properties

Label 2-2160-240.149-c0-0-7
Degree $2$
Conductor $2160$
Sign $-0.608 - 0.793i$
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  + (−0.258 − 0.965i)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 − 1.93·17-s + (−0.366 + 0.366i)19-s + (0.965 + 0.258i)20-s − 1.93i·23-s + 1.00i·25-s − 1.73·31-s + (−0.965 − 0.258i)32-s + (0.499 + 1.86i)34-s + (0.448 + 0.258i)38-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)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 − 1.93·17-s + (−0.366 + 0.366i)19-s + (0.965 + 0.258i)20-s − 1.93i·23-s + 1.00i·25-s − 1.73·31-s + (−0.965 − 0.258i)32-s + (0.499 + 1.86i)34-s + (0.448 + 0.258i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.608 - 0.793i$
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: \((2,\ 2160,\ (\ :0),\ -0.608 - 0.793i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1877772088\)
\(L(\frac12)\) \(\approx\) \(0.1877772088\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + T^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + 1.93T + T^{2} \)
19 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
23 \( 1 + 1.93iT - T^{2} \)
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 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
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

−8.687894005667826125572206753152, −8.432658945437036359953509387603, −7.42029000348678384467239691173, −6.50841802203037545644025924211, −5.22824164434273796825583794599, −4.42055326699267940563424323546, −3.93531829538049800467284191464, −2.70753123968109366687122760993, −1.70291367149656094349483665737, −0.13946743865191874362584295474, 1.92471852166543379741744696544, 3.40607197094563245814478633429, 4.18803425205742804271935829671, 5.07482965589896514407475055165, 6.02732242548462694280413575190, 6.90096662777749407372671659741, 7.24419114749213409134920593290, 8.146139713815036468085229734611, 8.843735818129565999537630856763, 9.535956104831781341166008958239

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