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Properties

Label 2-1680-7.2-c1-0-7
Degree $2$
Conductor $1680$
Sign $-0.155 - 0.987i$
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  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (−2.63 + 0.245i)7-s + (−0.499 + 0.866i)9-s + (−1.92 − 3.33i)11-s + 0.209·13-s + 0.999·15-s + (3.66 + 6.34i)17-s + (0.5 − 0.866i)19-s + (−1.52 − 2.15i)21-s + (−3.13 + 5.42i)23-s + (−0.499 − 0.866i)25-s − 0.999·27-s + 3.20·29-s + (5.23 + 9.07i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.223 − 0.387i)5-s + (−0.995 + 0.0927i)7-s + (−0.166 + 0.288i)9-s + (−0.580 − 1.00i)11-s + 0.0580·13-s + 0.258·15-s + (0.888 + 1.53i)17-s + (0.114 − 0.198i)19-s + (−0.333 − 0.471i)21-s + (−0.653 + 1.13i)23-s + (−0.0999 − 0.173i)25-s − 0.192·27-s + 0.595·29-s + (0.940 + 1.62i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.155 - 0.987i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ -0.155 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.282003381\)
\(L(\frac12)\) \(\approx\) \(1.282003381\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 - 0.245i)T \)
good11 \( 1 + (1.92 + 3.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.209T + 13T^{2} \)
17 \( 1 + (-3.66 - 6.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.13 - 5.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.20T + 29T^{2} \)
31 \( 1 + (-5.23 - 9.07i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.95 - 6.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.26T + 41T^{2} \)
43 \( 1 + 2.84T + 43T^{2} \)
47 \( 1 + (2.73 - 4.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.52 - 7.84i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.39 - 7.61i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.42 - 2.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 + (5.63 + 9.75i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.179 + 0.311i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.20T + 83T^{2} \)
89 \( 1 + (3.45 - 5.98i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−9.699891815025189627702847350461, −8.590569581515778709164756727173, −8.372091488934302871446924749902, −7.22682618721496326036817573795, −6.05605893332825445760503105206, −5.69085651456778617351526719646, −4.56187965960075036909377890283, −3.45820767401382408886681173336, −2.94626401559529630655921305362, −1.34553656938248704343014882161, 0.48088743018928932095811419482, 2.22837775637324665941174664900, 2.86462696011883405096737668018, 3.95466969700081031597230570870, 5.10250577106568618688785991673, 6.05435457397004477267599158098, 6.85900452145187793161710611783, 7.42002174491530076948101789093, 8.220608010166144566599136092002, 9.281425776148190646011415804204

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