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Properties

Label 2-2160-9.7-c1-0-20
Degree $2$
Conductor $2160$
Sign $0.224 + 0.974i$
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 + (2.04 − 3.53i)7-s + (0.675 − 1.17i)11-s + (−0.324 − 0.561i)13-s + 1.35·17-s − 0.648·19-s + (−2.39 − 4.14i)23-s + (−0.499 + 0.866i)25-s + (1.93 − 3.35i)29-s + (−3.84 − 6.66i)31-s + 4.08·35-s + 7.52·37-s + (−0.0898 − 0.155i)41-s + (−0.410 + 0.710i)43-s + (−5.45 + 9.44i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (0.772 − 1.33i)7-s + (0.203 − 0.353i)11-s + (−0.0898 − 0.155i)13-s + 0.327·17-s − 0.148·19-s + (−0.499 − 0.864i)23-s + (−0.0999 + 0.173i)25-s + (0.359 − 0.623i)29-s + (−0.691 − 1.19i)31-s + 0.690·35-s + 1.23·37-s + (−0.0140 − 0.0243i)41-s + (−0.0625 + 0.108i)43-s + (−0.795 + 1.37i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.224 + 0.974i$
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: \((2,\ 2160,\ (\ :1/2),\ 0.224 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.830402900\)
\(L(\frac12)\) \(\approx\) \(1.830402900\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.04 + 3.53i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.675 + 1.17i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.324 + 0.561i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.35T + 17T^{2} \)
19 \( 1 + 0.648T + 19T^{2} \)
23 \( 1 + (2.39 + 4.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.93 + 3.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.84 + 6.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.52T + 37T^{2} \)
41 \( 1 + (0.0898 + 0.155i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.410 - 0.710i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.45 - 9.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4.17T + 53T^{2} \)
59 \( 1 + (2.08 + 3.61i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.91 + 3.30i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.07 - 7.05i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.11T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + (-5.17 + 8.95i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.12 + 10.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (-6.79 + 11.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.881453051227863013701906930764, −7.81940123083778602255905365819, −7.63517372817096072669635701798, −6.51147125970406255213775624152, −5.90790732183762024986785719509, −4.68036972585571199175148210959, −4.12542621047640744614085196216, −3.07662195499823728094611048648, −1.87598392028204922930448567643, −0.65652598847265337500246717919, 1.45500261268612365252538727359, 2.24967196433754062005416952435, 3.40542483297605207004129960848, 4.61024263982754732496506591490, 5.27634553221676275244905001305, 5.91591676134434695114047314647, 6.89977500902218979614806116875, 7.85919692873619041641357922063, 8.536046637496299992420899834038, 9.157262310598355349537883076302

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