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Properties

Label 2-1680-3.2-c2-0-80
Degree $2$
Conductor $1680$
Sign $0.291 + 0.956i$
Analytic cond. $45.7766$
Root an. cond. $6.76584$
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  + (2.87 − 0.873i)3-s + 2.23i·5-s − 2.64·7-s + (7.47 − 5.01i)9-s − 6.37i·11-s + 8.60·13-s + (1.95 + 6.41i)15-s − 19.7i·17-s + 15.5·19-s + (−7.59 + 2.31i)21-s − 42.3i·23-s − 5.00·25-s + (17.0 − 20.9i)27-s + 46.9i·29-s − 56.3·31-s + ⋯
L(s)  = 1  + (0.956 − 0.291i)3-s + 0.447i·5-s − 0.377·7-s + (0.830 − 0.557i)9-s − 0.579i·11-s + 0.661·13-s + (0.130 + 0.427i)15-s − 1.16i·17-s + 0.816·19-s + (−0.361 + 0.110i)21-s − 1.84i·23-s − 0.200·25-s + (0.632 − 0.774i)27-s + 1.61i·29-s − 1.81·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.291 + 0.956i$
Analytic conductor: \(45.7766\)
Root analytic conductor: \(6.76584\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1),\ 0.291 + 0.956i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.662380105\)
\(L(\frac12)\) \(\approx\) \(2.662380105\)
\(L(2)\) 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 \( 1 + (-2.87 + 0.873i)T \)
5 \( 1 - 2.23iT \)
7 \( 1 + 2.64T \)
good11 \( 1 + 6.37iT - 121T^{2} \)
13 \( 1 - 8.60T + 169T^{2} \)
17 \( 1 + 19.7iT - 289T^{2} \)
19 \( 1 - 15.5T + 361T^{2} \)
23 \( 1 + 42.3iT - 529T^{2} \)
29 \( 1 - 46.9iT - 841T^{2} \)
31 \( 1 + 56.3T + 961T^{2} \)
37 \( 1 + 59.2T + 1.36e3T^{2} \)
41 \( 1 - 46.8iT - 1.68e3T^{2} \)
43 \( 1 - 75.1T + 1.84e3T^{2} \)
47 \( 1 + 43.9iT - 2.20e3T^{2} \)
53 \( 1 + 25.2iT - 2.80e3T^{2} \)
59 \( 1 + 40.0iT - 3.48e3T^{2} \)
61 \( 1 - 65.1T + 3.72e3T^{2} \)
67 \( 1 - 0.811T + 4.48e3T^{2} \)
71 \( 1 + 130. iT - 5.04e3T^{2} \)
73 \( 1 + 21.1T + 5.32e3T^{2} \)
79 \( 1 - 35.7T + 6.24e3T^{2} \)
83 \( 1 - 29.0iT - 6.88e3T^{2} \)
89 \( 1 + 82.1iT - 7.92e3T^{2} \)
97 \( 1 - 112.T + 9.40e3T^{2} \)
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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

−8.920370446993228714681023456596, −8.371670287761083759273264676181, −7.25050986595685101537124581361, −6.91131186184449183676893044459, −5.89184007426207955791574700455, −4.80542268627228809572317831650, −3.52372229516167982917244186241, −3.11742214533572623064734853011, −1.99303847282659771943572673271, −0.63875959987133670550806923061, 1.33521532402476483215794534204, 2.27149168864159624958821590826, 3.67093899579401491002552960004, 3.91978434492326529041381983453, 5.24237644668973872331639785751, 5.97461295884355922803633716238, 7.30723429866931472689964347264, 7.67749261006783842106126383978, 8.730553308298672044546006559336, 9.226392509163929571107876139104

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