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Properties

Label 2-1620-9.4-c3-0-15
Degree $2$
Conductor $1620$
Sign $-0.173 - 0.984i$
Analytic cond. $95.5830$
Root an. cond. $9.77666$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.5 − 4.33i)5-s + (9.95 + 17.2i)7-s + (16.9 + 29.3i)11-s + (3.95 − 6.85i)13-s + 12.9·17-s + 38.9·19-s + (−42.3 + 73.3i)23-s + (−12.5 − 21.6i)25-s + (51.7 + 89.7i)29-s + (−151. + 261. i)31-s + 99.5·35-s + 74·37-s + (56.6 − 98.1i)41-s + (−166. − 288. i)43-s + (121. + 210. i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (0.537 + 0.931i)7-s + (0.464 + 0.805i)11-s + (0.0844 − 0.146i)13-s + 0.184·17-s + 0.469·19-s + (−0.383 + 0.664i)23-s + (−0.100 − 0.173i)25-s + (0.331 + 0.574i)29-s + (−0.876 + 1.51i)31-s + 0.481·35-s + 0.328·37-s + (0.215 − 0.373i)41-s + (−0.591 − 1.02i)43-s + (0.377 + 0.653i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(95.5830\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :3/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.054140598\)
\(L(\frac12)\) \(\approx\) \(2.054140598\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.5 + 4.33i)T \)
good7 \( 1 + (-9.95 - 17.2i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-16.9 - 29.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-3.95 + 6.85i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 12.9T + 4.91e3T^{2} \)
19 \( 1 - 38.9T + 6.85e3T^{2} \)
23 \( 1 + (42.3 - 73.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-51.7 - 89.7i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (151. - 261. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 74T + 5.06e4T^{2} \)
41 \( 1 + (-56.6 + 98.1i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (166. + 288. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-121. - 210. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 447.T + 1.48e5T^{2} \)
59 \( 1 + (175. - 304. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (134. + 233. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (235. - 407. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 903.T + 3.57e5T^{2} \)
73 \( 1 + 80.3T + 3.89e5T^{2} \)
79 \( 1 + (326. + 565. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (62.3 + 108. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 149.T + 7.04e5T^{2} \)
97 \( 1 + (395. + 684. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.031225075933314297011496805872, −8.718497511116836539997439608865, −7.63912156354864975145361730336, −6.93957141340486876094548703673, −5.78237220079593002885345976732, −5.25782774775513622777465652611, −4.36168396348757629680304581334, −3.23034199951019992710276065845, −2.05416011373715669652022818335, −1.26342665359376549794664164560, 0.45493524988342680123375729753, 1.49484828778036034335357014782, 2.72513435934439476417132851569, 3.80299333410470938163756811699, 4.48036721781650494404838539015, 5.68083798218779670455200677967, 6.34842819879974967142664434395, 7.29403924427637589161496290095, 7.923679765437632512125913773613, 8.775517441364665137693603249745

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