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Properties

Label 2-1620-9.2-c2-0-27
Degree $2$
Conductor $1620$
Sign $-0.939 + 0.342i$
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  + (1.93 − 1.11i)5-s + (−2.74 + 4.75i)7-s + (−7.94 − 4.58i)11-s + (−5.74 − 9.94i)13-s + 16.9i·17-s + 26.9·19-s + (4.27 − 2.46i)23-s + (2.5 − 4.33i)25-s + (17.7 + 10.2i)29-s + (−10.4 − 18.1i)31-s + 12.2i·35-s − 62.4·37-s + (−35.4 + 20.4i)41-s + (−0.513 + 0.888i)43-s + (−74.6 − 43.1i)47-s + ⋯
L(s)  = 1  + (0.387 − 0.223i)5-s + (−0.391 + 0.678i)7-s + (−0.722 − 0.416i)11-s + (−0.441 − 0.765i)13-s + 0.998i·17-s + 1.41·19-s + (0.185 − 0.107i)23-s + (0.100 − 0.173i)25-s + (0.612 + 0.353i)29-s + (−0.338 − 0.585i)31-s + 0.350i·35-s − 1.68·37-s + (−0.864 + 0.499i)41-s + (−0.0119 + 0.0206i)43-s + (−1.58 − 0.917i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ -0.939 + 0.342i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3066002669\)
\(L(\frac12)\) \(\approx\) \(0.3066002669\)
\(L(2)\) 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 \( 1 + (2.74 - 4.75i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (7.94 + 4.58i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (5.74 + 9.94i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 16.9iT - 289T^{2} \)
19 \( 1 - 26.9T + 361T^{2} \)
23 \( 1 + (-4.27 + 2.46i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-17.7 - 10.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (10.4 + 18.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 62.4T + 1.36e3T^{2} \)
41 \( 1 + (35.4 - 20.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (0.513 - 0.888i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (74.6 + 43.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 96.0iT - 2.80e3T^{2} \)
59 \( 1 + (97.3 - 56.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-33.4 + 57.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-38 - 65.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 24.0iT - 5.04e3T^{2} \)
73 \( 1 + 18.9T + 5.32e3T^{2} \)
79 \( 1 + (53.4 - 92.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (39.1 + 22.5i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 115. iT - 7.92e3T^{2} \)
97 \( 1 + (-43.5 + 75.3i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−8.726986047715187260335594832649, −8.240954955293610990621236939711, −7.27796217945738637564705586121, −6.31370506962569315513439571475, −5.45353908301790296415222805804, −5.03156390661272990772411086798, −3.49818991654411969119937000733, −2.80161635534760173254375591955, −1.58514509015207029432679017934, −0.079689813120561900714951891034, 1.41242162094055244958471098050, 2.66855334073935105145791396721, 3.49685598436819181951404365539, 4.76090651955846028136300503834, 5.30527572510144795707203271279, 6.54359583928422648543065100825, 7.15844307927515240182494757009, 7.74162787387100647798520801534, 8.961122851103271601307607299215, 9.646294564354383122279965300680

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