Properties

Label 2-1620-3.2-c2-0-0
Degree $2$
Conductor $1620$
Sign $-1$
Analytic cond. $44.1418$
Root an. cond. $6.64392$
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  + 2.23i·5-s + 3.74·7-s − 11.6i·11-s − 2.25·13-s + 11.6i·17-s − 26.7·19-s + 19.9i·23-s − 5.00·25-s − 44.1i·29-s − 52.2·31-s + 8.37i·35-s + 14·37-s + 25.9i·41-s + 41.9·43-s + 45.4i·47-s + ⋯
L(s)  = 1  + 0.447i·5-s + 0.535·7-s − 1.06i·11-s − 0.173·13-s + 0.687i·17-s − 1.40·19-s + 0.865i·23-s − 0.200·25-s − 1.52i·29-s − 1.68·31-s + 0.239i·35-s + 0.378·37-s + 0.633i·41-s + 0.976·43-s + 0.967i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-1$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ -1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1169247418\)
\(L(\frac12)\) \(\approx\) \(0.1169247418\)
\(L(2)\) 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 \( 1 - 2.23iT \)
good7 \( 1 - 3.74T + 49T^{2} \)
11 \( 1 + 11.6iT - 121T^{2} \)
13 \( 1 + 2.25T + 169T^{2} \)
17 \( 1 - 11.6iT - 289T^{2} \)
19 \( 1 + 26.7T + 361T^{2} \)
23 \( 1 - 19.9iT - 529T^{2} \)
29 \( 1 + 44.1iT - 841T^{2} \)
31 \( 1 + 52.2T + 961T^{2} \)
37 \( 1 - 14T + 1.36e3T^{2} \)
41 \( 1 - 25.9iT - 1.68e3T^{2} \)
43 \( 1 - 41.9T + 1.84e3T^{2} \)
47 \( 1 - 45.4iT - 2.20e3T^{2} \)
53 \( 1 + 10.8iT - 2.80e3T^{2} \)
59 \( 1 - 36.8iT - 3.48e3T^{2} \)
61 \( 1 + 45.2T + 3.72e3T^{2} \)
67 \( 1 + 99.9T + 4.48e3T^{2} \)
71 \( 1 - 102. iT - 5.04e3T^{2} \)
73 \( 1 + 13.7T + 5.32e3T^{2} \)
79 \( 1 - 56.7T + 6.24e3T^{2} \)
83 \( 1 + 90.0iT - 6.88e3T^{2} \)
89 \( 1 + 95.2iT - 7.92e3T^{2} \)
97 \( 1 + 101.T + 9.40e3T^{2} \)
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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

−9.570308591617158058890465318476, −8.737892621673883834057318801494, −8.013187309427814171273683075675, −7.33969295558200096770668842584, −6.15970897227217964844375716738, −5.79656139501053034604940781705, −4.51238994888004061919462036592, −3.72608008771942564870850771616, −2.63325838304997810318208448354, −1.54367581354277065395592416331, 0.02959238095213150678776427720, 1.57622149625905186355325440316, 2.45180552415144701020216778696, 3.86783696303392165121042411670, 4.71660898823058885755283521418, 5.30093334597389427333472266525, 6.49182846621171582706329831500, 7.24776222347560542578288247748, 8.005106009938807893984758242220, 8.941309160216376065932687459455

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