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Properties

Label 2-1680-105.104-c1-0-32
Degree $2$
Conductor $1680$
Sign $0.951 - 0.306i$
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  + (0.726 − 1.57i)3-s + (−2.20 − 0.341i)5-s + (2.06 + 1.64i)7-s + (−1.94 − 2.28i)9-s + 1.06i·11-s + 4.82·13-s + (−2.14 + 3.22i)15-s + 7.89i·17-s + 4.02i·19-s + (4.09 − 2.05i)21-s − 5.69·23-s + (4.76 + 1.50i)25-s + (−5.00 + 1.40i)27-s + 2.00i·29-s + 4.89i·31-s + ⋯
L(s)  = 1  + (0.419 − 0.907i)3-s + (−0.988 − 0.152i)5-s + (0.781 + 0.623i)7-s + (−0.648 − 0.761i)9-s + 0.320i·11-s + 1.33·13-s + (−0.552 + 0.833i)15-s + 1.91i·17-s + 0.922i·19-s + (0.893 − 0.448i)21-s − 1.18·23-s + (0.953 + 0.301i)25-s + (−0.962 + 0.269i)27-s + 0.372i·29-s + 0.879i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.951 - 0.306i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ 0.951 - 0.306i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.678121683\)
\(L(\frac12)\) \(\approx\) \(1.678121683\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.726 + 1.57i)T \)
5 \( 1 + (2.20 + 0.341i)T \)
7 \( 1 + (-2.06 - 1.64i)T \)
good11 \( 1 - 1.06iT - 11T^{2} \)
13 \( 1 - 4.82T + 13T^{2} \)
17 \( 1 - 7.89iT - 17T^{2} \)
19 \( 1 - 4.02iT - 19T^{2} \)
23 \( 1 + 5.69T + 23T^{2} \)
29 \( 1 - 2.00iT - 29T^{2} \)
31 \( 1 - 4.89iT - 31T^{2} \)
37 \( 1 + 2.56iT - 37T^{2} \)
41 \( 1 - 5.08T + 41T^{2} \)
43 \( 1 + 6.15iT - 43T^{2} \)
47 \( 1 + 2.27iT - 47T^{2} \)
53 \( 1 - 9.84T + 53T^{2} \)
59 \( 1 - 5.87T + 59T^{2} \)
61 \( 1 + 7.02iT - 61T^{2} \)
67 \( 1 - 10.8iT - 67T^{2} \)
71 \( 1 - 0.0512iT - 71T^{2} \)
73 \( 1 - 2.86T + 73T^{2} \)
79 \( 1 + 7.00T + 79T^{2} \)
83 \( 1 - 7.59iT - 83T^{2} \)
89 \( 1 - 9.72T + 89T^{2} \)
97 \( 1 + 1.77T + 97T^{2} \)
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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

−8.899886467903731538697238767599, −8.386089782766806445606291342570, −8.077594853863893971644063204551, −7.16106794471506085899966939296, −6.16738943332648021844098832511, −5.53886210459146634098547599337, −4.06400077275930779643970606334, −3.59404542234487322120614942351, −2.11028062797709416505407724350, −1.27451444007005212168944854956, 0.68545419460543233061244413858, 2.52929929904799040252146132915, 3.51873541455725111661756495712, 4.27084829170905312606752292635, 4.86847670511882670556423744811, 5.98468809048489416168274410791, 7.19476160501911936438407453936, 7.85370167862990795506817879488, 8.497949145425789391163246511930, 9.227352070334261193703384831026

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