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Properties

Label 2-1680-420.419-c0-0-15
Degree $2$
Conductor $1680$
Sign $i$
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  + (0.866 − 0.5i)3-s − i·5-s − i·7-s + (0.499 − 0.866i)9-s + 11-s − 1.73·13-s + (−0.5 − 0.866i)15-s + i·17-s + (−0.5 − 0.866i)21-s − 25-s − 0.999i·27-s + 1.73i·29-s + (0.866 − 0.5i)33-s − 35-s + (−1.49 + 0.866i)39-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s − i·5-s − i·7-s + (0.499 − 0.866i)9-s + 11-s − 1.73·13-s + (−0.5 − 0.866i)15-s + i·17-s + (−0.5 − 0.866i)21-s − 25-s − 0.999i·27-s + 1.73i·29-s + (0.866 − 0.5i)33-s − 35-s + (−1.49 + 0.866i)39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (1679, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.464572803\)
\(L(\frac12)\) \(\approx\) \(1.464572803\)
\(L(1)\) 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 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + iT \)
7 \( 1 + iT \)
good11 \( 1 - T + T^{2} \)
13 \( 1 + 1.73T + T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 1.73iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 1.73T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - 2T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 1.73iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - 1.73T + T^{2} \)
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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

−9.229028683063855995939169804733, −8.618138186881685144929528748733, −7.70426314625899119938688782130, −7.18330872677057291742920661314, −6.33931212191465120786657419134, −5.06157997532744247146879330368, −4.22360472209068158954983967112, −3.48822044193045791330754972479, −2.08345048530866697880619083467, −1.09376431497562220319225998027, 2.28083746152587388766368859299, 2.62519508762557178782060566754, 3.74689896726053221120384154096, 4.68787943899462933867607517997, 5.63761617239729834831469811923, 6.71300171560713561425891611623, 7.41688567664293489434690064089, 8.147037680718704388085924399422, 9.246735080310201504563860117661, 9.563445464306919519647566738948

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