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Properties

Label 2-2160-12.11-c1-0-10
Degree $2$
Conductor $2160$
Sign $0.5 - 0.866i$
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  + i·5-s − 2.70i·7-s − 0.972·11-s + 13-s + 7.68i·17-s + 2.70i·19-s − 0.972·23-s − 25-s − 1.68i·29-s + 0.972i·31-s + 2.70·35-s − 2.68·37-s + 6i·41-s + 0.972i·43-s + 11.3·47-s + ⋯
L(s)  = 1  + 0.447i·5-s − 1.02i·7-s − 0.293·11-s + 0.277·13-s + 1.86i·17-s + 0.620i·19-s − 0.202·23-s − 0.200·25-s − 0.312i·29-s + 0.174i·31-s + 0.457·35-s − 0.441·37-s + 0.937i·41-s + 0.148i·43-s + 1.65·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.5 - 0.866i$
Analytic conductor: \(17.2476\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ 0.5 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.505007595\)
\(L(\frac12)\) \(\approx\) \(1.505007595\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good7 \( 1 + 2.70iT - 7T^{2} \)
11 \( 1 + 0.972T + 11T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 - 7.68iT - 17T^{2} \)
19 \( 1 - 2.70iT - 19T^{2} \)
23 \( 1 + 0.972T + 23T^{2} \)
29 \( 1 + 1.68iT - 29T^{2} \)
31 \( 1 - 0.972iT - 31T^{2} \)
37 \( 1 + 2.68T + 37T^{2} \)
41 \( 1 - 6iT - 41T^{2} \)
43 \( 1 - 0.972iT - 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 1.94T + 59T^{2} \)
61 \( 1 - 6.68T + 61T^{2} \)
67 \( 1 + 11.5iT - 67T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 - 8.68T + 73T^{2} \)
79 \( 1 - 14.0iT - 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 - 12iT - 89T^{2} \)
97 \( 1 + 6.68T + 97T^{2} \)
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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

−9.258029285017206990753041936992, −8.155567836246304480042999234975, −7.83311337405910668874865786660, −6.78563703889401615615673227318, −6.20833462817058715802677197562, −5.28269046297040398850126954908, −4.02084505005763448291661651020, −3.69956153251853148014139674587, −2.34552357118876088639618397039, −1.16350646010946162937767706556, 0.58634757129867134202025140046, 2.14847246988671341060390936098, 2.91157625172917142156538653987, 4.10453424469558777576131361202, 5.22577802362742036090487473938, 5.46453264772581385371863096506, 6.66456237424959839897484505324, 7.37749996684709179546081111296, 8.319743127016927451636045062688, 9.027597732273530017236466297714

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