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Properties

Label 2-2160-15.2-c1-0-47
Degree $2$
Conductor $2160$
Sign $-0.989 - 0.146i$
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  + (−0.189 − 2.22i)5-s + (−3.07 − 3.07i)7-s − 5.65i·11-s + (3.45 − 3.45i)13-s + (−3.31 + 3.31i)17-s − 3.45i·19-s + (1.88 + 1.88i)23-s + (−4.92 + 0.843i)25-s + 0.117·29-s − 2.82·31-s + (−6.27 + 7.43i)35-s + (7.06 + 7.06i)37-s − 3.09i·41-s + (3.92 − 3.92i)43-s + (−2.88 + 2.88i)47-s + ⋯
L(s)  = 1  + (−0.0846 − 0.996i)5-s + (−1.16 − 1.16i)7-s − 1.70i·11-s + (0.958 − 0.958i)13-s + (−0.803 + 0.803i)17-s − 0.792i·19-s + (0.393 + 0.393i)23-s + (−0.985 + 0.168i)25-s + 0.0217·29-s − 0.508·31-s + (−1.06 + 1.25i)35-s + (1.16 + 1.16i)37-s − 0.482i·41-s + (0.597 − 0.597i)43-s + (−0.421 + 0.421i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.989 - 0.146i$
Analytic conductor: \(17.2476\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ -0.989 - 0.146i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9851811497\)
\(L(\frac12)\) \(\approx\) \(0.9851811497\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.189 + 2.22i)T \)
good7 \( 1 + (3.07 + 3.07i)T + 7iT^{2} \)
11 \( 1 + 5.65iT - 11T^{2} \)
13 \( 1 + (-3.45 + 3.45i)T - 13iT^{2} \)
17 \( 1 + (3.31 - 3.31i)T - 17iT^{2} \)
19 \( 1 + 3.45iT - 19T^{2} \)
23 \( 1 + (-1.88 - 1.88i)T + 23iT^{2} \)
29 \( 1 - 0.117T + 29T^{2} \)
31 \( 1 + 2.82T + 31T^{2} \)
37 \( 1 + (-7.06 - 7.06i)T + 37iT^{2} \)
41 \( 1 + 3.09iT - 41T^{2} \)
43 \( 1 + (-3.92 + 3.92i)T - 43iT^{2} \)
47 \( 1 + (2.88 - 2.88i)T - 47iT^{2} \)
53 \( 1 + (-3.00 - 3.00i)T + 53iT^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 - 3.08T + 61T^{2} \)
67 \( 1 + (-3.45 - 3.45i)T + 67iT^{2} \)
71 \( 1 - 9.07iT - 71T^{2} \)
73 \( 1 + (-9.37 + 9.37i)T - 73iT^{2} \)
79 \( 1 - 5.85iT - 79T^{2} \)
83 \( 1 + (11.0 + 11.0i)T + 83iT^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 + (-1.54 - 1.54i)T + 97iT^{2} \)
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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

−8.645389829501341823204082872394, −8.086118421943429718974832228727, −7.09551401270880813388340262524, −6.15060498194219269368664920076, −5.70895869486968121201208017992, −4.46855643058303023649949879205, −3.67473950446518944832557532661, −3.02193940146926525831309379167, −1.14845039223669357479202056561, −0.37877620742744164645882778581, 1.99162199934795862295701881032, 2.64266568788863543796546037353, 3.68349330895747631425570585633, 4.55402704750988257675390578007, 5.75780068730172071722717403677, 6.50174917776254431339240984328, 6.91951664978203999614226217772, 7.81386757045016481875678530366, 8.980016230903052393444374293991, 9.459781456383198747444892659021

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