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Properties

Label 2-1680-28.19-c1-0-8
Degree $2$
Conductor $1680$
Sign $0.832 - 0.553i$
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 + (−0.866 − 0.5i)5-s + (−1.73 − 2i)7-s + (−0.499 + 0.866i)9-s + (1.5 − 0.866i)11-s + 4.46i·13-s + 0.999i·15-s + (−4.73 + 2.73i)17-s + (−1.23 + 2.13i)19-s + (−0.866 + 2.5i)21-s + (3.86 + 2.23i)23-s + (0.499 + 0.866i)25-s + 0.999·27-s − 2·29-s + (3 + 5.19i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.387 − 0.223i)5-s + (−0.654 − 0.755i)7-s + (−0.166 + 0.288i)9-s + (0.452 − 0.261i)11-s + 1.23i·13-s + 0.258i·15-s + (−1.14 + 0.662i)17-s + (−0.282 + 0.489i)19-s + (−0.188 + 0.545i)21-s + (0.806 + 0.465i)23-s + (0.0999 + 0.173i)25-s + 0.192·27-s − 0.371·29-s + (0.538 + 0.933i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ 0.832 - 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9729879365\)
\(L(\frac12)\) \(\approx\) \(0.9729879365\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (1.73 + 2i)T \)
good11 \( 1 + (-1.5 + 0.866i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.46iT - 13T^{2} \)
17 \( 1 + (4.73 - 2.73i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.23 - 2.13i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.86 - 2.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.33 + 5.76i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.267iT - 41T^{2} \)
43 \( 1 + 5.46iT - 43T^{2} \)
47 \( 1 + (-5.59 + 9.69i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.86 - 4.96i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.46 - 6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.73 - 4.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.19 - 1.26i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 4iT - 71T^{2} \)
73 \( 1 + (11.6 - 6.73i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-11.1 - 6.46i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 14T + 83T^{2} \)
89 \( 1 + (-10.3 - 6i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 17.8iT - 97T^{2} \)
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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

−9.164649475142665828528158055326, −8.802489767531913782596210967390, −7.69506755391170055699312693741, −6.86017675180146514295519355828, −6.51827845705329438146401543052, −5.41940354655659880458486421052, −4.21940930779181614319457106452, −3.73909051532974112648665464222, −2.24883281491600921063023348861, −1.01928643341640543824087984438, 0.46843952113676642144552491851, 2.49215161338570423180073755847, 3.20547420097953342321972571232, 4.35705986951188133277168730155, 5.09327605413716588322271610867, 6.13803719870783677074006068852, 6.69022766937522328109618165477, 7.71873646043824477991052316029, 8.646741938568246401490610591429, 9.321425659562599025539328112698

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