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Properties

Label 2-1620-45.29-c2-0-26
Degree $2$
Conductor $1620$
Sign $0.593 + 0.804i$
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  + (−3.62 − 3.44i)5-s + (5.87 + 3.39i)7-s + (−8.57 − 4.94i)11-s + (−17.6 + 10.1i)13-s + 19.1·17-s + 12·19-s + (−4.79 − 8.30i)23-s + (1.24 + 24.9i)25-s + (7.34 + 4.24i)29-s + (19 + 32.9i)31-s + (−9.59 − 32.5i)35-s − 6.78i·37-s + (60.0 − 34.6i)41-s + (−58.7 − 33.9i)43-s + (38.3 − 66.4i)47-s + ⋯
L(s)  = 1  + (−0.724 − 0.689i)5-s + (0.839 + 0.484i)7-s + (−0.779 − 0.449i)11-s + (−1.35 + 0.782i)13-s + 1.12·17-s + 0.631·19-s + (−0.208 − 0.361i)23-s + (0.0498 + 0.998i)25-s + (0.253 + 0.146i)29-s + (0.612 + 1.06i)31-s + (−0.274 − 0.929i)35-s − 0.183i·37-s + (1.46 − 0.845i)41-s + (−1.36 − 0.788i)43-s + (0.816 − 1.41i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 + 0.804i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.593 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.593 + 0.804i$
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: \((2,\ 1620,\ (\ :1),\ 0.593 + 0.804i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.504660810\)
\(L(\frac12)\) \(\approx\) \(1.504660810\)
\(L(2)\) 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 \( 1 + (3.62 + 3.44i)T \)
good7 \( 1 + (-5.87 - 3.39i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (8.57 + 4.94i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (17.6 - 10.1i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 19.1T + 289T^{2} \)
19 \( 1 - 12T + 361T^{2} \)
23 \( 1 + (4.79 + 8.30i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-7.34 - 4.24i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-19 - 32.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 6.78iT - 1.36e3T^{2} \)
41 \( 1 + (-60.0 + 34.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (58.7 + 33.9i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-38.3 + 66.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 + (72.2 - 41.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-35 + 60.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-93.9 + 54.2i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 118. iT - 5.04e3T^{2} \)
73 \( 1 + 13.5iT - 5.32e3T^{2} \)
79 \( 1 + (15 - 25.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-67.1 + 116. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 32.5iT - 7.92e3T^{2} \)
97 \( 1 + (-82.2 - 47.4i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−8.945810387214120066101112105304, −8.244395030511101446227771967953, −7.66944661640880212131480657829, −6.88391657616458122633416092789, −5.34660605123647730052283378188, −5.18235138789848826801424613842, −4.15446739787947228112294900717, −3.01307147942563627788342620934, −1.87519522960586846762291479927, −0.53563941042292312692510881476, 0.881142415728753404656334120808, 2.45062412640099605973355419350, 3.22434597582822635688556808585, 4.42543287984913743799415506876, 5.04684864348409257699211566322, 6.07430073501712310669721942705, 7.29009803483618674586991515902, 7.86861546995827703704317573626, 7.940999351646838783831397387111, 9.632430287586188979970949577496

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