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Properties

Label 2-1620-1620.979-c0-0-0
Degree $2$
Conductor $1620$
Sign $-0.790 - 0.612i$
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  + (0.597 + 0.802i)2-s + (0.893 + 0.448i)3-s + (−0.286 + 0.957i)4-s + (−0.835 + 0.549i)5-s + (0.173 + 0.984i)6-s + (0.0798 − 0.0845i)7-s + (−0.939 + 0.342i)8-s + (0.597 + 0.802i)9-s + (−0.939 − 0.342i)10-s + (−0.686 + 0.727i)12-s + (0.115 + 0.0135i)14-s + (−0.993 + 0.116i)15-s + (−0.835 − 0.549i)16-s + (−0.286 + 0.957i)18-s + (−0.286 − 0.957i)20-s + (0.109 − 0.0397i)21-s + ⋯
L(s)  = 1  + (0.597 + 0.802i)2-s + (0.893 + 0.448i)3-s + (−0.286 + 0.957i)4-s + (−0.835 + 0.549i)5-s + (0.173 + 0.984i)6-s + (0.0798 − 0.0845i)7-s + (−0.939 + 0.342i)8-s + (0.597 + 0.802i)9-s + (−0.939 − 0.342i)10-s + (−0.686 + 0.727i)12-s + (0.115 + 0.0135i)14-s + (−0.993 + 0.116i)15-s + (−0.835 − 0.549i)16-s + (−0.286 + 0.957i)18-s + (−0.286 − 0.957i)20-s + (0.109 − 0.0397i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.790 - 0.612i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ -0.790 - 0.612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.654496495\)
\(L(\frac12)\) \(\approx\) \(1.654496495\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.597 - 0.802i)T \)
3 \( 1 + (-0.893 - 0.448i)T \)
5 \( 1 + (0.835 - 0.549i)T \)
good7 \( 1 + (-0.0798 + 0.0845i)T + (-0.0581 - 0.998i)T^{2} \)
11 \( 1 + (-0.597 + 0.802i)T^{2} \)
13 \( 1 + (0.686 - 0.727i)T^{2} \)
17 \( 1 + (-0.173 + 0.984i)T^{2} \)
19 \( 1 + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (-0.941 - 0.998i)T + (-0.0581 + 0.998i)T^{2} \)
29 \( 1 + (1.93 - 0.225i)T + (0.973 - 0.230i)T^{2} \)
31 \( 1 + (-0.893 + 0.448i)T^{2} \)
37 \( 1 + (0.939 - 0.342i)T^{2} \)
41 \( 1 + (-0.473 + 0.635i)T + (-0.286 - 0.957i)T^{2} \)
43 \( 1 + (-0.109 + 1.87i)T + (-0.993 - 0.116i)T^{2} \)
47 \( 1 + (-1.16 - 0.275i)T + (0.893 + 0.448i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.597 - 0.802i)T^{2} \)
61 \( 1 + (0.512 + 1.71i)T + (-0.835 + 0.549i)T^{2} \)
67 \( 1 + (-1.97 - 0.230i)T + (0.973 + 0.230i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-0.766 + 0.642i)T^{2} \)
79 \( 1 + (0.286 - 0.957i)T^{2} \)
83 \( 1 + (-0.473 - 0.635i)T + (-0.286 + 0.957i)T^{2} \)
89 \( 1 + (1.12 - 0.408i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (-0.396 - 0.918i)T^{2} \)
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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

−9.571574671542455516364435061650, −8.954730742083777416021422936907, −8.143249841491063318164072749192, −7.36574307003351365572472196581, −7.08659214781533850246538286538, −5.73793500019226342230825119313, −4.87398953401884830423088259775, −3.84364056192509123155752047934, −3.47824823381794499381507015505, −2.34415409650363142519678841769, 1.03229455790525522683899251337, 2.26343276940070835628274811903, 3.24943994612016545147605402034, 4.05364642066810197625864499393, 4.80444389029006834066846111277, 5.89960260303806982026225720337, 6.96137540118603283713759360083, 7.76107384576194159622447676391, 8.678185574663712512080032928065, 9.176113244196944438818881859098

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