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Properties

Label 2-1620-45.29-c2-0-19
Degree $2$
Conductor $1620$
Sign $-0.0249 - 0.999i$
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  + (2.22 + 4.47i)5-s + (4.03 + 2.32i)7-s + (10.4 + 6.01i)11-s + (−7.33 + 4.23i)13-s + 1.25·17-s + 16.4·19-s + (1.19 + 2.06i)23-s + (−15.0 + 19.9i)25-s + (20.7 + 11.9i)29-s + (2.62 + 4.55i)31-s + (−1.44 + 23.2i)35-s + 25.1i·37-s + (25.8 − 14.9i)41-s + (22.0 + 12.7i)43-s + (32.4 − 56.2i)47-s + ⋯
L(s)  = 1  + (0.445 + 0.895i)5-s + (0.576 + 0.332i)7-s + (0.946 + 0.546i)11-s + (−0.564 + 0.325i)13-s + 0.0736·17-s + 0.867·19-s + (0.0519 + 0.0899i)23-s + (−0.603 + 0.797i)25-s + (0.716 + 0.413i)29-s + (0.0847 + 0.146i)31-s + (−0.0413 + 0.663i)35-s + 0.680i·37-s + (0.630 − 0.363i)41-s + (0.513 + 0.296i)43-s + (0.691 − 1.19i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0249 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0249 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.0249 - 0.999i$
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: \((2,\ 1620,\ (\ :1),\ -0.0249 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.406315434\)
\(L(\frac12)\) \(\approx\) \(2.406315434\)
\(L(2)\) 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 \( 1 + (-2.22 - 4.47i)T \)
good7 \( 1 + (-4.03 - 2.32i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-10.4 - 6.01i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (7.33 - 4.23i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 1.25T + 289T^{2} \)
19 \( 1 - 16.4T + 361T^{2} \)
23 \( 1 + (-1.19 - 2.06i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-20.7 - 11.9i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-2.62 - 4.55i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 25.1iT - 1.36e3T^{2} \)
41 \( 1 + (-25.8 + 14.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-22.0 - 12.7i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-32.4 + 56.2i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 71.4T + 2.80e3T^{2} \)
59 \( 1 + (16.0 - 9.25i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (9.55 - 16.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (70.2 - 40.5i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 21.9iT - 5.04e3T^{2} \)
73 \( 1 - 109. iT - 5.32e3T^{2} \)
79 \( 1 + (-67.0 + 116. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-7.80 + 13.5i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 79.4iT - 7.92e3T^{2} \)
97 \( 1 + (-112. - 65.1i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.480037435306843028505889456817, −8.737909189210407796786109633667, −7.64192055692675824058996745302, −7.02018285142020743984580684306, −6.27121919559546833244448636627, −5.33117044978498292691426099879, −4.45324411439265669645603654309, −3.33643213427696819023231301325, −2.34321759042953910023332747239, −1.36581238520518264061884788981, 0.69467495898659628148092367603, 1.56275043647633805216659423909, 2.87944156568349023744153832995, 4.12349099808655404886153455675, 4.82298879271956352610098270280, 5.70613739155304798223041765496, 6.46000270509233071593757556852, 7.64103405269743558837373939776, 8.130714595738127816774263439414, 9.252227361004903052885276781988

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