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Properties

Label 2-1680-21.20-c1-0-31
Degree $2$
Conductor $1680$
Sign $0.885 - 0.464i$
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  + (1.68 + 0.396i)3-s − 5-s + (2 − 1.73i)7-s + (2.68 + 1.33i)9-s − 0.792i·11-s + 5.84i·13-s + (−1.68 − 0.396i)15-s + 1.37·17-s + 3.46i·19-s + (4.05 − 2.12i)21-s − 1.87i·23-s + 25-s + (4 + 3.31i)27-s + 4.25i·29-s − 3.46i·31-s + ⋯
L(s)  = 1  + (0.973 + 0.228i)3-s − 0.447·5-s + (0.755 − 0.654i)7-s + (0.895 + 0.445i)9-s − 0.238i·11-s + 1.61i·13-s + (−0.435 − 0.102i)15-s + 0.332·17-s + 0.794i·19-s + (0.885 − 0.464i)21-s − 0.391i·23-s + 0.200·25-s + (0.769 + 0.638i)27-s + 0.790i·29-s − 0.622i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.464i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.885 - 0.464i$
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: \((2,\ 1680,\ (\ :1/2),\ 0.885 - 0.464i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.564117408\)
\(L(\frac12)\) \(\approx\) \(2.564117408\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 0.396i)T \)
5 \( 1 + T \)
7 \( 1 + (-2 + 1.73i)T \)
good11 \( 1 + 0.792iT - 11T^{2} \)
13 \( 1 - 5.84iT - 13T^{2} \)
17 \( 1 - 1.37T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + 1.87iT - 23T^{2} \)
29 \( 1 - 4.25iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 - 4.74T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 6.74T + 43T^{2} \)
47 \( 1 + 7.37T + 47T^{2} \)
53 \( 1 + 8.51iT - 53T^{2} \)
59 \( 1 - 2.74T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 - 6.74T + 67T^{2} \)
71 \( 1 + 13.5iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + 3.37T + 79T^{2} \)
83 \( 1 - 5.48T + 83T^{2} \)
89 \( 1 + 3.25T + 89T^{2} \)
97 \( 1 + 1.08iT - 97T^{2} \)
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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

−9.322826848263723321705192912304, −8.572519343639229973987428763006, −7.87585502731024658073639570439, −7.28095670863277969052689940965, −6.39047849515602000675173100642, −5.01461464765252755683756380081, −4.19445178964892010985379073208, −3.68684771420567822690251204346, −2.36666545389775700404331736834, −1.32455728090469331706002060625, 1.02956962566077506643294647505, 2.40408100984573914011962450512, 3.10279823236284401344635616593, 4.20316566587963827219290240667, 5.11403700186365082839247634254, 6.04151724150269393732651940750, 7.24127039420798067517899482969, 7.86830949402985419538663305564, 8.324731416056718536894088310854, 9.175087440465228558693154404424

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