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Properties

Label 2-1620-12.11-c1-0-41
Degree $2$
Conductor $1620$
Sign $-0.696 - 0.717i$
Analytic cond. $12.9357$
Root an. cond. $3.59663$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.531 + 1.31i)2-s + (−1.43 + 1.39i)4-s + i·5-s + 1.38i·7-s + (−2.58 − 1.13i)8-s + (−1.31 + 0.531i)10-s + 6.29·11-s + 5.01·13-s + (−1.81 + 0.734i)14-s + (0.117 − 3.99i)16-s + 1.58i·17-s + 0.313i·19-s + (−1.39 − 1.43i)20-s + (3.34 + 8.24i)22-s − 0.478·23-s + ⋯
L(s)  = 1  + (0.375 + 0.926i)2-s + (−0.717 + 0.696i)4-s + 0.447i·5-s + 0.522i·7-s + (−0.915 − 0.402i)8-s + (−0.414 + 0.168i)10-s + 1.89·11-s + 1.38·13-s + (−0.484 + 0.196i)14-s + (0.0294 − 0.999i)16-s + 0.384i·17-s + 0.0720i·19-s + (−0.311 − 0.320i)20-s + (0.713 + 1.75i)22-s − 0.0997·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.696 - 0.717i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ -0.696 - 0.717i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.160559865\)
\(L(\frac12)\) \(\approx\) \(2.160559865\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.531 - 1.31i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 - 1.38iT - 7T^{2} \)
11 \( 1 - 6.29T + 11T^{2} \)
13 \( 1 - 5.01T + 13T^{2} \)
17 \( 1 - 1.58iT - 17T^{2} \)
19 \( 1 - 0.313iT - 19T^{2} \)
23 \( 1 + 0.478T + 23T^{2} \)
29 \( 1 + 1.71iT - 29T^{2} \)
31 \( 1 - 9.47iT - 31T^{2} \)
37 \( 1 - 8.55T + 37T^{2} \)
41 \( 1 - 7.54iT - 41T^{2} \)
43 \( 1 + 6.02iT - 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 + 8.86iT - 53T^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
61 \( 1 + 5.02T + 61T^{2} \)
67 \( 1 - 3.38iT - 67T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + 4.55T + 73T^{2} \)
79 \( 1 + 2.85iT - 79T^{2} \)
83 \( 1 - 0.983T + 83T^{2} \)
89 \( 1 + 2.64iT - 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−9.320093816053233546686017116577, −8.809140342844507532962877687935, −8.130930062862058506926959478451, −7.07549630985018072664263779511, −6.23958333521639260930126151806, −6.07566033189843740283738193691, −4.73644185425878423027949187762, −3.84237181229308376877637866320, −3.16359207355721008425135664925, −1.43342238214648008058172746028, 0.875168968231543383006777347774, 1.67374012269572618125160426207, 3.18087324260690337493202875529, 4.08154114811253239357323584382, 4.48760447622816668155108618487, 5.91475255141497299146826457022, 6.31507066068405293880586648925, 7.57763081525843260878859277193, 8.677492073264023942317256843854, 9.208797821800074954454854331455

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