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Properties

Label 2-1680-420.299-c0-0-2
Degree $2$
Conductor $1680$
Sign $0.386 + 0.922i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
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  + (−0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (0.499 + 0.866i)15-s + 0.999·21-s + (−1.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + 0.999·27-s − 1.73i·29-s − 0.999·35-s − 41-s + 43-s − 0.999·45-s + (1 − 1.73i)47-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (0.499 + 0.866i)15-s + 0.999·21-s + (−1.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + 0.999·27-s − 1.73i·29-s − 0.999·35-s − 41-s + 43-s − 0.999·45-s + (1 − 1.73i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :0),\ 0.386 + 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8026271634\)
\(L(\frac12)\) \(\approx\) \(0.8026271634\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.641347458820670140129695224302, −8.733421593440622122598923852390, −7.995276860042532827846135007568, −6.78048211973294559108674657388, −6.03058359976064822080661548707, −5.31851390437344165795213164239, −4.25571127787601886633561105733, −3.88560023290614734310380781288, −2.35372106524720229646721077826, −0.64735347805078364783007446092, 1.71109909061342186585175870631, 2.57857537076480085720570055899, 3.54518282677485937360527143267, 5.13155357403236123285393777816, 5.85218478210296263593321147965, 6.40552936112820986540451153791, 7.17937600976479477381049315522, 7.943020544878074736382515926524, 8.923225139944419663497769973559, 9.719787224032579981702215013698

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