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Properties

Label 2-1680-21.20-c1-0-53
Degree $2$
Conductor $1680$
Sign $-0.940 + 0.340i$
Analytic cond. $13.4148$
Root an. cond. $3.66263$
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  + (0.227 − 1.71i)3-s − 5-s + (1.22 + 2.34i)7-s + (−2.89 − 0.781i)9-s − 4.73i·11-s − 5.25i·13-s + (−0.227 + 1.71i)15-s + 0.432·17-s + 6.30i·19-s + (4.30 − 1.56i)21-s − 0.332i·23-s + 25-s + (−2.00 + 4.79i)27-s − 7.48i·29-s + 0.0758i·31-s + ⋯
L(s)  = 1  + (0.131 − 0.991i)3-s − 0.447·5-s + (0.461 + 0.887i)7-s + (−0.965 − 0.260i)9-s − 1.42i·11-s − 1.45i·13-s + (−0.0587 + 0.443i)15-s + 0.104·17-s + 1.44i·19-s + (0.940 − 0.340i)21-s − 0.0692i·23-s + 0.200·25-s + (−0.385 + 0.922i)27-s − 1.39i·29-s + 0.0136i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.940 + 0.340i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ -0.940 + 0.340i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.020510085\)
\(L(\frac12)\) \(\approx\) \(1.020510085\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.227 + 1.71i)T \)
5 \( 1 + T \)
7 \( 1 + (-1.22 - 2.34i)T \)
good11 \( 1 + 4.73iT - 11T^{2} \)
13 \( 1 + 5.25iT - 13T^{2} \)
17 \( 1 - 0.432T + 17T^{2} \)
19 \( 1 - 6.30iT - 19T^{2} \)
23 \( 1 + 0.332iT - 23T^{2} \)
29 \( 1 + 7.48iT - 29T^{2} \)
31 \( 1 - 0.0758iT - 31T^{2} \)
37 \( 1 + 2.54T + 37T^{2} \)
41 \( 1 + 4.82T + 41T^{2} \)
43 \( 1 + 1.97T + 43T^{2} \)
47 \( 1 + 9.28T + 47T^{2} \)
53 \( 1 + 13.8iT - 53T^{2} \)
59 \( 1 - 5.65T + 59T^{2} \)
61 \( 1 + 5.11iT - 61T^{2} \)
67 \( 1 + 6.75T + 67T^{2} \)
71 \( 1 + 9.35iT - 71T^{2} \)
73 \( 1 + 2.75iT - 73T^{2} \)
79 \( 1 + 0.0508T + 79T^{2} \)
83 \( 1 + 4.12T + 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + 11.1iT - 97T^{2} \)
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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

−8.462686547140251538651341863250, −8.279783214277530368278232336873, −7.72865363574118504731980333953, −6.47032708700945611537928108276, −5.80019342548009042076950652348, −5.21547767975236880529945153848, −3.58175965924648052445001904539, −2.93091913396337149875108199362, −1.71455055750520950848123230528, −0.37512735807395934397342421468, 1.65537765008266962180932692575, 2.97104469621635394369939234042, 4.17408941689936786269544695498, 4.50044316363876591592150408755, 5.26437808316479532177151370006, 6.89900787833167970662461639421, 7.09265940338081971925027849683, 8.261983849394304333535643317807, 9.033809995815135909880113985956, 9.681073427112678324511000768100

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