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Properties

Label 2-2160-15.14-c2-0-6
Degree $2$
Conductor $2160$
Sign $-0.707 - 0.707i$
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  + (3.53 − 3.53i)5-s + 5i·7-s − 1.41i·11-s − 9i·13-s − 11.3·17-s − 21·19-s + 1.41·23-s − 25.0i·25-s + 38.1i·29-s − 40·31-s + (17.6 + 17.6i)35-s + 25i·37-s + 52.3i·41-s + 64i·43-s + 22.6·47-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)5-s + 0.714i·7-s − 0.128i·11-s − 0.692i·13-s − 0.665·17-s − 1.10·19-s + 0.0614·23-s − 1.00i·25-s + 1.31i·29-s − 1.29·31-s + (0.505 + 0.505i)35-s + 0.675i·37-s + 1.27i·41-s + 1.48i·43-s + 0.481·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.707 - 0.707i$
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: \((2,\ 2160,\ (\ :1),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5954977003\)
\(L(\frac12)\) \(\approx\) \(0.5954977003\)
\(L(2)\) 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 \( 1 + (-3.53 + 3.53i)T \)
good7 \( 1 - 5iT - 49T^{2} \)
11 \( 1 + 1.41iT - 121T^{2} \)
13 \( 1 + 9iT - 169T^{2} \)
17 \( 1 + 11.3T + 289T^{2} \)
19 \( 1 + 21T + 361T^{2} \)
23 \( 1 - 1.41T + 529T^{2} \)
29 \( 1 - 38.1iT - 841T^{2} \)
31 \( 1 + 40T + 961T^{2} \)
37 \( 1 - 25iT - 1.36e3T^{2} \)
41 \( 1 - 52.3iT - 1.68e3T^{2} \)
43 \( 1 - 64iT - 1.84e3T^{2} \)
47 \( 1 - 22.6T + 2.20e3T^{2} \)
53 \( 1 + 72.1T + 2.80e3T^{2} \)
59 \( 1 + 90.5iT - 3.48e3T^{2} \)
61 \( 1 + 97T + 3.72e3T^{2} \)
67 \( 1 + 131iT - 4.48e3T^{2} \)
71 \( 1 - 89.0iT - 5.04e3T^{2} \)
73 \( 1 - 17iT - 5.32e3T^{2} \)
79 \( 1 - 117T + 6.24e3T^{2} \)
83 \( 1 + 57.9T + 6.88e3T^{2} \)
89 \( 1 - 147. iT - 7.92e3T^{2} \)
97 \( 1 - 41iT - 9.40e3T^{2} \)
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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

−9.176410235074926734081731739691, −8.499458238294057053171123929856, −7.86074355653614687308563263222, −6.60076542143041766899026620553, −6.06068095944580790446690847324, −5.17080828204824696250574661241, −4.58682408313203652006964305614, −3.28914110157508641836244195694, −2.29652701738692474314126576570, −1.35324049645282745087552969738, 0.13508166798981376824091877532, 1.76894622033294934605330827258, 2.46469806440888987275187681810, 3.77160875468792456922303715010, 4.38001022824450552800813770398, 5.58456596733105015064515099083, 6.30668597144511230388065753821, 7.06496771434930337393498562400, 7.57683409115659473416068047037, 8.852388189139691522678511801231

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