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Properties

Label 2-2160-9.2-c2-0-17
Degree $2$
Conductor $2160$
Sign $-0.342 - 0.939i$
Analytic cond. $58.8557$
Root an. cond. $7.67174$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.93 + 1.11i)5-s + (−5.87 + 10.1i)7-s + (13.1 + 7.57i)11-s + (8.87 + 15.3i)13-s − 15.1i·17-s + 11.2·19-s + (29.2 − 16.8i)23-s + (2.5 − 4.33i)25-s + (−8.23 − 4.75i)29-s + (28.1 + 48.6i)31-s − 26.2i·35-s + 14·37-s + (22.5 − 12.9i)41-s + (−9.99 + 17.3i)43-s + (62.6 + 36.1i)47-s + ⋯
L(s)  = 1  + (−0.387 + 0.223i)5-s + (−0.838 + 1.45i)7-s + (1.19 + 0.688i)11-s + (0.682 + 1.18i)13-s − 0.891i·17-s + 0.592·19-s + (1.27 − 0.733i)23-s + (0.100 − 0.173i)25-s + (−0.284 − 0.164i)29-s + (0.906 + 1.57i)31-s − 0.750i·35-s + 0.378·37-s + (0.548 − 0.316i)41-s + (−0.232 + 0.402i)43-s + (1.33 + 0.769i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(58.8557\)
Root analytic conductor: \(7.67174\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1),\ -0.342 - 0.939i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.911784037\)
\(L(\frac12)\) \(\approx\) \(1.911784037\)
\(L(2)\) 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 \( 1 + (1.93 - 1.11i)T \)
good7 \( 1 + (5.87 - 10.1i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-13.1 - 7.57i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-8.87 - 15.3i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 15.1iT - 289T^{2} \)
19 \( 1 - 11.2T + 361T^{2} \)
23 \( 1 + (-29.2 + 16.8i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (8.23 + 4.75i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-28.1 - 48.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 14T + 1.36e3T^{2} \)
41 \( 1 + (-22.5 + 12.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (9.99 - 17.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-62.6 - 36.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 37.6iT - 2.80e3T^{2} \)
59 \( 1 + (55.1 - 31.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (0.618 - 1.07i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-42.9 - 74.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 22.1iT - 5.04e3T^{2} \)
73 \( 1 + 60.2T + 5.32e3T^{2} \)
79 \( 1 + (-51.6 + 89.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (78 + 45.0i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 12.0iT - 7.92e3T^{2} \)
97 \( 1 + (49.8 - 86.3i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−9.070987516444723734152399590674, −8.731117279391778808029654904573, −7.40579587854973827090713999654, −6.66583538966855005041215071839, −6.23815531205974755219894868323, −5.10938578662391252640132466635, −4.26676860046853419802175643567, −3.22910799283999924091780783616, −2.46351873647662955124014954851, −1.17707553305712283958066325555, 0.60777361732385959559167709955, 1.18156326639398668372176884846, 3.11702518747926252298455180037, 3.68675188433559511462929770764, 4.32392219935286735565107297109, 5.66108020847162495790946867432, 6.29994376629366258866840677016, 7.16979348368167677186954834400, 7.80905166839430325862377247309, 8.642891298667950451470598136360

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