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Properties

Label 2-2160-9.4-c1-0-22
Degree $2$
Conductor $2160$
Sign $-0.870 - 0.492i$
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 + (−2.62 − 4.54i)7-s + (−1.33 − 2.31i)11-s + (−1.90 + 3.30i)13-s − 3.52·17-s + 4.67·19-s + (2.47 − 4.29i)23-s + (−0.499 − 0.866i)25-s + (−0.928 − 1.60i)29-s + (−4.33 + 7.51i)31-s − 5.24·35-s − 2.67·37-s + (−1.83 + 3.18i)41-s + (1.76 + 3.05i)43-s + (−4.63 − 8.02i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.991 − 1.71i)7-s + (−0.402 − 0.697i)11-s + (−0.529 + 0.916i)13-s − 0.855·17-s + 1.07·19-s + (0.516 − 0.895i)23-s + (−0.0999 − 0.173i)25-s + (−0.172 − 0.298i)29-s + (−0.778 + 1.34i)31-s − 0.886·35-s − 0.439·37-s + (−0.286 + 0.496i)41-s + (0.269 + 0.466i)43-s + (−0.675 − 1.17i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.870 - 0.492i$
Analytic conductor: \(17.2476\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ -0.870 - 0.492i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3082793654\)
\(L(\frac12)\) \(\approx\) \(0.3082793654\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 + 4.54i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.33 + 2.31i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.90 - 3.30i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.52T + 17T^{2} \)
19 \( 1 - 4.67T + 19T^{2} \)
23 \( 1 + (-2.47 + 4.29i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.928 + 1.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.33 - 7.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.67T + 37T^{2} \)
41 \( 1 + (1.83 - 3.18i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.76 - 3.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.63 + 8.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.85T + 53T^{2} \)
59 \( 1 + (2.10 - 3.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.98 - 6.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.429 - 0.744i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.1T + 71T^{2} \)
73 \( 1 - 6.28T + 73T^{2} \)
79 \( 1 + (-2.81 - 4.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.94 - 3.37i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11T + 89T^{2} \)
97 \( 1 + (-1.91 - 3.32i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.770614872609000176510382934421, −7.72719738391955879087353575882, −6.92093438787723889235668068717, −6.56939517212247228368447472614, −5.35186140097574436596925756478, −4.50062557008012117682668512328, −3.70198910923628510213242779956, −2.78768429184516881543897874330, −1.28855464790656788236911916305, −0.10699063553636920540899910273, 2.03074875807476523847717325945, 2.76387268896428630962040964021, 3.51674813748847389114060644558, 5.05566109955793437751633733703, 5.53158524356384511322657168422, 6.31317140050395627143718965432, 7.20262944607450193540203403683, 7.903331344372968422354611643948, 9.036238584334048634450069573074, 9.430870048309043528695124133642

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