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Properties

Label 2-1620-1620.439-c0-0-1
Degree $2$
Conductor $1620$
Sign $-0.135 + 0.990i$
Analytic cond. $0.808485$
Root an. cond. $0.899158$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.396 + 0.918i)2-s + (0.835 − 0.549i)3-s + (−0.686 − 0.727i)4-s + (−0.0581 − 0.998i)5-s + (0.173 + 0.984i)6-s + (−1.73 − 0.412i)7-s + (0.939 − 0.342i)8-s + (0.396 − 0.918i)9-s + (0.939 + 0.342i)10-s + (−0.973 − 0.230i)12-s + (1.06 − 1.43i)14-s + (−0.597 − 0.802i)15-s + (−0.0581 + 0.998i)16-s + (0.686 + 0.727i)18-s + (−0.686 + 0.727i)20-s + (−1.67 + 0.611i)21-s + ⋯
L(s)  = 1  + (−0.396 + 0.918i)2-s + (0.835 − 0.549i)3-s + (−0.686 − 0.727i)4-s + (−0.0581 − 0.998i)5-s + (0.173 + 0.984i)6-s + (−1.73 − 0.412i)7-s + (0.939 − 0.342i)8-s + (0.396 − 0.918i)9-s + (0.939 + 0.342i)10-s + (−0.973 − 0.230i)12-s + (1.06 − 1.43i)14-s + (−0.597 − 0.802i)15-s + (−0.0581 + 0.998i)16-s + (0.686 + 0.727i)18-s + (−0.686 + 0.727i)20-s + (−1.67 + 0.611i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.135 + 0.990i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ -0.135 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6800840740\)
\(L(\frac12)\) \(\approx\) \(0.6800840740\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.396 - 0.918i)T \)
3 \( 1 + (-0.835 + 0.549i)T \)
5 \( 1 + (0.0581 + 0.998i)T \)
good7 \( 1 + (1.73 + 0.412i)T + (0.893 + 0.448i)T^{2} \)
11 \( 1 + (-0.396 - 0.918i)T^{2} \)
13 \( 1 + (-0.973 - 0.230i)T^{2} \)
17 \( 1 + (-0.173 + 0.984i)T^{2} \)
19 \( 1 + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (1.89 - 0.448i)T + (0.893 - 0.448i)T^{2} \)
29 \( 1 + (0.342 + 0.460i)T + (-0.286 + 0.957i)T^{2} \)
31 \( 1 + (0.835 + 0.549i)T^{2} \)
37 \( 1 + (0.939 - 0.342i)T^{2} \)
41 \( 1 + (0.786 + 1.82i)T + (-0.686 + 0.727i)T^{2} \)
43 \( 1 + (-1.67 + 0.843i)T + (0.597 - 0.802i)T^{2} \)
47 \( 1 + (-0.227 - 0.758i)T + (-0.835 + 0.549i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.396 + 0.918i)T^{2} \)
61 \( 1 + (-1.14 + 1.21i)T + (-0.0581 - 0.998i)T^{2} \)
67 \( 1 + (0.713 - 0.957i)T + (-0.286 - 0.957i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-0.766 + 0.642i)T^{2} \)
79 \( 1 + (0.686 + 0.727i)T^{2} \)
83 \( 1 + (-0.786 + 1.82i)T + (-0.686 - 0.727i)T^{2} \)
89 \( 1 + (0.744 - 0.270i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (0.993 + 0.116i)T^{2} \)
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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

−9.215691481955253819861627995166, −8.602021005228203301291764821971, −7.71225649741207319979720572282, −7.16704221036387940441247462954, −6.21645474815146110012138829253, −5.67630030167568181719753000639, −4.18284557782662600405010752793, −3.63575282931473179534756930656, −2.03362698173260713541803911400, −0.52294765925618949563436484099, 2.17946760855932627085347205112, 2.89545795511909943186900608559, 3.55065007143411207481708106702, 4.28433032423058277084008792459, 5.78105174518309387696033399648, 6.73211856412749749584869680349, 7.65056774232671220987737945213, 8.437838990118403053764526006358, 9.322327355902492026853014004157, 9.876700118309521007258298323240

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