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Properties

Label 2-1680-28.19-c1-0-12
Degree $2$
Conductor $1680$
Sign $-0.495 - 0.868i$
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 + (2.43 + 1.03i)7-s + (−0.499 + 0.866i)9-s + (−1.99 + 1.15i)11-s + 1.49i·13-s + 0.999i·15-s + (−6.54 + 3.77i)17-s + (−0.534 + 0.924i)19-s + (0.321 + 2.62i)21-s + (−0.292 − 0.168i)23-s + (0.499 + 0.866i)25-s − 0.999·27-s − 3.13·29-s + (2.96 + 5.14i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.387 + 0.223i)5-s + (0.920 + 0.391i)7-s + (−0.166 + 0.288i)9-s + (−0.601 + 0.347i)11-s + 0.414i·13-s + 0.258i·15-s + (−1.58 + 0.916i)17-s + (−0.122 + 0.212i)19-s + (0.0700 + 0.573i)21-s + (−0.0609 − 0.0352i)23-s + (0.0999 + 0.173i)25-s − 0.192·27-s − 0.582·29-s + (0.533 + 0.923i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.495 - 0.868i$
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: \((2,\ 1680,\ (\ :1/2),\ -0.495 - 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.731548508\)
\(L(\frac12)\) \(\approx\) \(1.731548508\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.43 - 1.03i)T \)
good11 \( 1 + (1.99 - 1.15i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.49iT - 13T^{2} \)
17 \( 1 + (6.54 - 3.77i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.534 - 0.924i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.292 + 0.168i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.13T + 29T^{2} \)
31 \( 1 + (-2.96 - 5.14i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.97 - 3.41i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.54iT - 41T^{2} \)
43 \( 1 + 6.64iT - 43T^{2} \)
47 \( 1 + (-4.03 + 6.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.64 - 4.57i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.04 - 3.53i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.92 + 2.26i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.37 + 4.83i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.444iT - 71T^{2} \)
73 \( 1 + (1.32 - 0.763i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.67 - 2.70i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.130T + 83T^{2} \)
89 \( 1 + (-6.84 - 3.95i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.39iT - 97T^{2} \)
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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

−9.556390454689008863238178415389, −8.730420603277492149507876759832, −8.300191626738616703496056423393, −7.25640474347204674189227385376, −6.38884847643726994256974971967, −5.39624864874312207887979170265, −4.67959359487334047325787133855, −3.82878757433552501723809608050, −2.48383390382920187222954862848, −1.80788744280844325148454830369, 0.59882889643297953663384613658, 1.96720900163837413197534823197, 2.76481354853165349167850471403, 4.15171418377426924521099691296, 4.97063061107455079563179003137, 5.82549585531245977630180321725, 6.80712664100099100043992821812, 7.58246133061543439436504541857, 8.252388576729954228571088056371, 8.979670647387445327564044865034

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