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Properties

Label 2-1620-45.29-c2-0-21
Degree $2$
Conductor $1620$
Sign $0.578 - 0.815i$
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  + (4.91 + 0.897i)5-s + (10.9 + 6.33i)7-s + (−16.1 − 9.35i)11-s + (−5.88 + 3.39i)13-s − 11.5·17-s + 24.2·19-s + (14.6 + 25.3i)23-s + (23.3 + 8.82i)25-s + (12.9 + 7.47i)29-s + (−6.91 − 11.9i)31-s + (48.2 + 40.9i)35-s + 38.2i·37-s + (26.7 − 15.4i)41-s + (11.3 + 6.53i)43-s + (2.84 − 4.92i)47-s + ⋯
L(s)  = 1  + (0.983 + 0.179i)5-s + (1.56 + 0.904i)7-s + (−1.47 − 0.850i)11-s + (−0.452 + 0.261i)13-s − 0.679·17-s + 1.27·19-s + (0.635 + 1.10i)23-s + (0.935 + 0.353i)25-s + (0.446 + 0.257i)29-s + (−0.223 − 0.386i)31-s + (1.37 + 1.17i)35-s + 1.03i·37-s + (0.653 − 0.377i)41-s + (0.263 + 0.152i)43-s + (0.0605 − 0.104i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.578 - 0.815i$
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: \((2,\ 1620,\ (\ :1),\ 0.578 - 0.815i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.704579509\)
\(L(\frac12)\) \(\approx\) \(2.704579509\)
\(L(2)\) 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 \( 1 + (-4.91 - 0.897i)T \)
good7 \( 1 + (-10.9 - 6.33i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (16.1 + 9.35i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (5.88 - 3.39i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 11.5T + 289T^{2} \)
19 \( 1 - 24.2T + 361T^{2} \)
23 \( 1 + (-14.6 - 25.3i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-12.9 - 7.47i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (6.91 + 11.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 38.2iT - 1.36e3T^{2} \)
41 \( 1 + (-26.7 + 15.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-11.3 - 6.53i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-2.84 + 4.92i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 57.7T + 2.80e3T^{2} \)
59 \( 1 + (26.2 - 15.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (38.5 - 66.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-111. + 64.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 35.8iT - 5.04e3T^{2} \)
73 \( 1 + 40.6iT - 5.32e3T^{2} \)
79 \( 1 + (70.0 - 121. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (59.4 - 102. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 75.0iT - 7.92e3T^{2} \)
97 \( 1 + (73.1 + 42.2i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.260576920539325048702041627799, −8.550763508295686556451252635354, −7.83186391087950049281938179447, −7.02145079123335107905187623641, −5.59986217089276260936147859671, −5.49349997027527324675308573877, −4.64682177047938257703097488291, −2.96821322042342734997743135078, −2.31449924150558553136614678384, −1.24368715033352874009199759200, 0.76979038364227787396067049214, 1.94072618912097582168367769256, 2.71794156615269216250384472540, 4.38615477693974304316339526314, 4.97405861825506911801216524523, 5.51785953343521733103336290028, 6.87807429799218640317324581555, 7.52944203522564979335912834209, 8.171988412588338075042498122234, 9.113537478477244841647404728945

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