Properties

Label 2-2160-9.7-c1-0-14
Degree $2$
Conductor $2160$
Sign $0.993 - 0.113i$
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  + (0.5 + 0.866i)5-s + (−0.257 + 0.445i)7-s + (1.66 − 2.87i)11-s + (0.660 + 1.14i)13-s + 3.32·17-s + 1.32·19-s + (−2.06 − 3.57i)23-s + (−0.499 + 0.866i)25-s + (−0.693 + 1.20i)29-s + (4.36 + 7.56i)31-s − 0.514·35-s + 0.292·37-s + (−5.67 − 9.82i)41-s + (5.17 − 8.96i)43-s + (2.43 − 4.21i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (−0.0971 + 0.168i)7-s + (0.500 − 0.867i)11-s + (0.183 + 0.317i)13-s + 0.805·17-s + 0.303·19-s + (−0.430 − 0.745i)23-s + (−0.0999 + 0.173i)25-s + (−0.128 + 0.222i)29-s + (0.784 + 1.35i)31-s − 0.0869·35-s + 0.0481·37-s + (−0.886 − 1.53i)41-s + (0.789 − 1.36i)43-s + (0.354 − 0.614i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.993 - 0.113i$
Analytic conductor: \(17.2476\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ 0.993 - 0.113i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.954387839\)
\(L(\frac12)\) \(\approx\) \(1.954387839\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (0.257 - 0.445i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.66 + 2.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.660 - 1.14i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.32T + 17T^{2} \)
19 \( 1 - 1.32T + 19T^{2} \)
23 \( 1 + (2.06 + 3.57i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.693 - 1.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.36 - 7.56i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.292T + 37T^{2} \)
41 \( 1 + (5.67 + 9.82i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.17 + 8.96i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.43 + 4.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.02T + 53T^{2} \)
59 \( 1 + (-2.51 - 4.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.67 - 6.36i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.72 - 8.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.99T + 71T^{2} \)
73 \( 1 - 6.05T + 73T^{2} \)
79 \( 1 + (4.02 - 6.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.771 - 1.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (-6.12 + 10.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.821211676901633917419956792306, −8.619118497647194899591610632009, −7.41240954466065875644819312816, −6.76430970835077973234870543589, −5.91422348797827630940217889312, −5.29133809827497813880185146995, −4.04087265273792406440849049794, −3.30230326026974698682299541180, −2.27393770937549571756096317698, −0.947605974315996345337441939954, 0.963100331043213417544742957623, 2.06976605548305452036098238280, 3.29491012775657887481051555553, 4.21575174727123900955563335306, 5.04211816932934228162713306209, 5.93795969986625200380068730347, 6.64447886771378856887419553958, 7.73172852299930716543679194131, 8.065478071000301106920579486134, 9.334594132064341875956761970086

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