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Properties

Label 2-2160-15.14-c2-0-85
Degree $2$
Conductor $2160$
Sign $-0.354 + 0.934i$
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  + (4.67 + 1.77i)5-s − 9.67i·7-s − 8.58i·11-s − 19.0i·13-s + 26.2·17-s − 26.6·19-s + 25.8·23-s + (18.7 + 16.5i)25-s − 18.3i·29-s + 7.56·31-s + (17.1 − 45.2i)35-s + 47.0i·37-s + 0.602i·41-s − 62.2i·43-s − 79.2·47-s + ⋯
L(s)  = 1  + (0.934 + 0.354i)5-s − 1.38i·7-s − 0.780i·11-s − 1.46i·13-s + 1.54·17-s − 1.40·19-s + 1.12·23-s + (0.748 + 0.663i)25-s − 0.633i·29-s + 0.243·31-s + (0.490 − 1.29i)35-s + 1.27i·37-s + 0.0146i·41-s − 1.44i·43-s − 1.68·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 + 0.934i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.354 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.354 + 0.934i$
Analytic conductor: \(58.8557\)
Root analytic conductor: \(7.67174\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1),\ -0.354 + 0.934i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.195524475\)
\(L(\frac12)\) \(\approx\) \(2.195524475\)
\(L(2)\) 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 \( 1 + (-4.67 - 1.77i)T \)
good7 \( 1 + 9.67iT - 49T^{2} \)
11 \( 1 + 8.58iT - 121T^{2} \)
13 \( 1 + 19.0iT - 169T^{2} \)
17 \( 1 - 26.2T + 289T^{2} \)
19 \( 1 + 26.6T + 361T^{2} \)
23 \( 1 - 25.8T + 529T^{2} \)
29 \( 1 + 18.3iT - 841T^{2} \)
31 \( 1 - 7.56T + 961T^{2} \)
37 \( 1 - 47.0iT - 1.36e3T^{2} \)
41 \( 1 - 0.602iT - 1.68e3T^{2} \)
43 \( 1 + 62.2iT - 1.84e3T^{2} \)
47 \( 1 + 79.2T + 2.20e3T^{2} \)
53 \( 1 + 80.9T + 2.80e3T^{2} \)
59 \( 1 + 15.4iT - 3.48e3T^{2} \)
61 \( 1 + 62.7T + 3.72e3T^{2} \)
67 \( 1 + 48.3iT - 4.48e3T^{2} \)
71 \( 1 - 99.6iT - 5.04e3T^{2} \)
73 \( 1 - 60.7iT - 5.32e3T^{2} \)
79 \( 1 + 60.8T + 6.24e3T^{2} \)
83 \( 1 - 77.0T + 6.88e3T^{2} \)
89 \( 1 - 28.7iT - 7.92e3T^{2} \)
97 \( 1 - 23.3iT - 9.40e3T^{2} \)
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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

−8.497795735563035068758299825249, −7.926277741319408507865010052865, −7.05374297536912331215295027737, −6.29313803773487121451208911586, −5.56235424960251835500338101774, −4.70402660745777426230087945774, −3.46242385943666980694305466929, −2.94778550327228898118700052001, −1.45565562617958540916285393484, −0.53342363632614774282972574390, 1.49874903102563990594638261815, 2.14048435696822535540036752973, 3.14377283839421280705011034777, 4.57618315042937533531264563492, 5.09288662030142537174071134969, 6.08744625893815798150342659796, 6.52657430694128270226452804667, 7.62038508662471202750183131588, 8.610852055610776960990192007497, 9.217288083025497852584186351115

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