Properties

Degree $2$
Conductor $1680$
Sign $0.605 - 0.795i$
Motivic weight $1$
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 + (−2 − 1.73i)7-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s + 7·13-s − 0.999·15-s + (2 + 3.46i)17-s + (0.5 − 0.866i)19-s + (0.499 − 2.59i)21-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s − 0.999·27-s − 8·29-s + (3 + 5.19i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.223 + 0.387i)5-s + (−0.755 − 0.654i)7-s + (−0.166 + 0.288i)9-s + (−0.150 − 0.261i)11-s + 1.94·13-s − 0.258·15-s + (0.485 + 0.840i)17-s + (0.114 − 0.198i)19-s + (0.109 − 0.566i)21-s + (0.104 − 0.180i)23-s + (−0.0999 − 0.173i)25-s − 0.192·27-s − 1.48·29-s + (0.538 + 0.933i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.605 - 0.795i$
Motivic weight: \(1\)
Character: $\chi_{1680} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.737288012\)
\(L(\frac12)\) \(\approx\) \(1.737288012\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2 + 1.73i)T \)
good11 \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 7T + 13T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.5 + 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14T + 71T^{2} \)
73 \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 16T + 97T^{2} \)
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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

−9.480497401198091138442506620414, −8.687309517365665087144115677869, −7.992089930100106946891583691678, −7.07737911629179315651648548378, −6.20620881299514860608967374891, −5.54673462980279822339860205102, −3.98760570273129490453574010863, −3.77103974420012038448572195734, −2.74554594674542759618937612781, −1.10077133267677927312309517926, 0.796668093031243616817377443772, 2.11437616740694529608020639293, 3.25994944754442350787060715362, 3.97882251719869522592019268861, 5.36131440446448228118158327346, 6.01366793770760016084888204001, 6.79885697797360055294478179234, 7.81427208947639451250651817126, 8.366840344029233311283527653321, 9.326571567935796344720809965382

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