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Properties

Label 2-1680-35.4-c1-0-4
Degree $2$
Conductor $1680$
Sign $-0.897 - 0.441i$
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  + (0.866 + 0.5i)3-s + (−1.86 − 1.23i)5-s + (1.73 + 2i)7-s + (0.499 + 0.866i)9-s + (−2.5 + 4.33i)11-s + i·13-s + (−1 − 2i)15-s + (−1.73 − i)17-s + (−3.5 − 6.06i)19-s + (0.499 + 2.59i)21-s + (−2.59 + 1.5i)23-s + (1.96 + 4.59i)25-s + 0.999i·27-s + (−3 + 5.19i)31-s + (−4.33 + 2.5i)33-s + ⋯
L(s)  = 1  + (0.499 + 0.288i)3-s + (−0.834 − 0.550i)5-s + (0.654 + 0.755i)7-s + (0.166 + 0.288i)9-s + (−0.753 + 1.30i)11-s + 0.277i·13-s + (−0.258 − 0.516i)15-s + (−0.420 − 0.242i)17-s + (−0.802 − 1.39i)19-s + (0.109 + 0.566i)21-s + (−0.541 + 0.312i)23-s + (0.392 + 0.919i)25-s + 0.192i·27-s + (−0.538 + 0.933i)31-s + (−0.753 + 0.435i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.897 - 0.441i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ -0.897 - 0.441i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7432604123\)
\(L(\frac12)\) \(\approx\) \(0.7432604123\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (1.86 + 1.23i)T \)
7 \( 1 + (-1.73 - 2i)T \)
good11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.59 - 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.33 + 2.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 + (11.2 - 6.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.866 - 0.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.19 + 3i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (3.46 + 2i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7 - 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10iT - 83T^{2} \)
89 \( 1 + (-5 - 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−9.423277817465540032354621130240, −8.884863865857028537872998909190, −8.190586521233070695174596934039, −7.48794799585419355844455250280, −6.70036382886893314687653294752, −5.16822638616059567750405277658, −4.82084159735300823595300041783, −3.96888670374430314587109230220, −2.67582961653597843891877120540, −1.80674691080792076270632642620, 0.24887481811153605779030336358, 1.82066084015965360069579524556, 3.09659087153969776061909428397, 3.78579758902243307153773159391, 4.66863527161857467515690422104, 5.94769890439093373656069368336, 6.66812792004240194767648781543, 7.82368365384378837104120755483, 8.013832820940433720434349385891, 8.620434413420405143045093869614

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