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Properties

Label 2-1620-45.29-c2-0-33
Degree $2$
Conductor $1620$
Sign $0.903 + 0.427i$
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  + (3.55 − 3.51i)5-s + (5.42 + 3.13i)7-s + (0.895 + 0.516i)11-s + (−2.59 + 1.5i)13-s + 15.7·17-s + 18.7·19-s + (−17.3 − 30.0i)23-s + (0.273 − 24.9i)25-s + (2.68 + 1.55i)29-s + (8.89 + 15.4i)31-s + (30.2 − 7.93i)35-s + 43.3i·37-s + (40.0 − 23.1i)41-s + (17.9 + 10.3i)43-s + (−39.4 + 68.3i)47-s + ⋯
L(s)  = 1  + (0.710 − 0.703i)5-s + (0.774 + 0.447i)7-s + (0.0813 + 0.0469i)11-s + (−0.199 + 0.115i)13-s + 0.928·17-s + 0.988·19-s + (−0.753 − 1.30i)23-s + (0.0109 − 0.999i)25-s + (0.0926 + 0.0534i)29-s + (0.286 + 0.496i)31-s + (0.865 − 0.226i)35-s + 1.17i·37-s + (0.977 − 0.564i)41-s + (0.417 + 0.241i)43-s + (−0.839 + 1.45i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.427i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.903 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.903 + 0.427i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ 0.903 + 0.427i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.740137084\)
\(L(\frac12)\) \(\approx\) \(2.740137084\)
\(L(2)\) 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 \( 1 + (-3.55 + 3.51i)T \)
good7 \( 1 + (-5.42 - 3.13i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-0.895 - 0.516i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (2.59 - 1.5i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 15.7T + 289T^{2} \)
19 \( 1 - 18.7T + 361T^{2} \)
23 \( 1 + (17.3 + 30.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-2.68 - 1.55i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-8.89 - 15.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 43.3iT - 1.36e3T^{2} \)
41 \( 1 + (-40.0 + 23.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-17.9 - 10.3i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (39.4 - 68.3i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 44.2T + 2.80e3T^{2} \)
59 \( 1 + (-78.4 + 45.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (4.39 - 7.60i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (16.7 - 9.65i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 56.6iT - 5.04e3T^{2} \)
73 \( 1 + 109. iT - 5.32e3T^{2} \)
79 \( 1 + (19.5 - 33.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (37.8 - 65.6i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 90.5iT - 7.92e3T^{2} \)
97 \( 1 + (-7.69 - 4.44i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.167103258102763580459531336695, −8.337329481336991941053582883504, −7.80127884927008729635188099472, −6.62284202052757869126007761693, −5.76073853162361593090704754699, −5.07059132300044476100393991045, −4.34646578205162226535548782248, −2.95030149465418723627902152319, −1.90282124700077097608546189839, −0.911261188969173784396267807265, 1.06846558687673032033137807185, 2.12198698300184624647168391871, 3.23267372886558495424443026221, 4.15397711718946665865767338383, 5.44090899932376737323834354785, 5.77709007846793957066437718282, 7.07225749832121119843378645837, 7.52626208144908214302343142754, 8.359911706514048085270712704125, 9.579674301801319710701274206245

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