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Properties

Label 2-1620-9.5-c2-0-18
Degree $2$
Conductor $1620$
Sign $0.996 + 0.0871i$
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 + (−1.24 − 2.16i)7-s + (14.5 − 8.39i)11-s + (−2.96 + 5.14i)13-s + 10.4i·17-s + 6.65·19-s + (9.29 + 5.36i)23-s + (2.5 + 4.33i)25-s + (−4.98 + 2.87i)29-s + (9.19 − 15.9i)31-s − 5.58i·35-s − 2.16·37-s + (−45.8 − 26.4i)41-s + (40.7 + 70.6i)43-s + (26.7 − 15.4i)47-s + ⋯
L(s)  = 1  + (0.387 + 0.223i)5-s + (−0.178 − 0.309i)7-s + (1.32 − 0.763i)11-s + (−0.228 + 0.395i)13-s + 0.617i·17-s + 0.350·19-s + (0.404 + 0.233i)23-s + (0.100 + 0.173i)25-s + (−0.171 + 0.0991i)29-s + (0.296 − 0.513i)31-s − 0.159i·35-s − 0.0583·37-s + (−1.11 − 0.645i)41-s + (0.948 + 1.64i)43-s + (0.569 − 0.328i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.996 + 0.0871i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ 0.996 + 0.0871i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.336151050\)
\(L(\frac12)\) \(\approx\) \(2.336151050\)
\(L(2)\) 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 \( 1 + (1.24 + 2.16i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-14.5 + 8.39i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (2.96 - 5.14i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 10.4iT - 289T^{2} \)
19 \( 1 - 6.65T + 361T^{2} \)
23 \( 1 + (-9.29 - 5.36i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (4.98 - 2.87i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-9.19 + 15.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 2.16T + 1.36e3T^{2} \)
41 \( 1 + (45.8 + 26.4i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-40.7 - 70.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-26.7 + 15.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 13.5iT - 2.80e3T^{2} \)
59 \( 1 + (23.5 + 13.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (24.4 + 42.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-31.2 + 54.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 44.7iT - 5.04e3T^{2} \)
73 \( 1 - 138.T + 5.32e3T^{2} \)
79 \( 1 + (2.58 + 4.47i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-36.5 + 21.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 89.6iT - 7.92e3T^{2} \)
97 \( 1 + (-13.5 - 23.5i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.264133167308712244350066871175, −8.519206015393320149292074030035, −7.54201546532289056237848111523, −6.63232907906746005056068566502, −6.14998187694608878232953325980, −5.11706899141159673223024485741, −3.99614666669627179557209284315, −3.30779757575942231428147986922, −1.99045127596418263011424337991, −0.865182094925677165071468856960, 0.908532951802346311599906950199, 2.06399441671643032040405639628, 3.15148551329188160104028770137, 4.25981797916512205688203219978, 5.09641040372361088034111144306, 5.97965854160081716300424806353, 6.84566388981696739462933386149, 7.48492110426979030281401281919, 8.659540941995169887777398488924, 9.223180832461588584824336034059

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