Properties

Degree $2$
Conductor $1680$
Sign $-0.975 - 0.218i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.618 − 1.61i)3-s − 5-s + (−0.381 − 2.61i)7-s + (−2.23 − 2.00i)9-s − 4.47i·11-s + 3.23i·13-s + (−0.618 + 1.61i)15-s − 0.763·17-s + 0.472i·19-s + (−4.47 − 1.00i)21-s − 4i·23-s + 25-s + (−4.61 + 2.38i)27-s − 5.70i·29-s + 7.23i·31-s + ⋯
L(s)  = 1  + (0.356 − 0.934i)3-s − 0.447·5-s + (−0.144 − 0.989i)7-s + (−0.745 − 0.666i)9-s − 1.34i·11-s + 0.897i·13-s + (−0.159 + 0.417i)15-s − 0.185·17-s + 0.108i·19-s + (−0.975 − 0.218i)21-s − 0.834i·23-s + 0.200·25-s + (−0.888 + 0.458i)27-s − 1.05i·29-s + 1.29i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.975 - 0.218i$
Motivic weight: \(1\)
Character: $\chi_{1680} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ -0.975 - 0.218i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9374554259\)
\(L(\frac12)\) \(\approx\) \(0.9374554259\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.618 + 1.61i)T \)
5 \( 1 + T \)
7 \( 1 + (0.381 + 2.61i)T \)
good11 \( 1 + 4.47iT - 11T^{2} \)
13 \( 1 - 3.23iT - 13T^{2} \)
17 \( 1 + 0.763T + 17T^{2} \)
19 \( 1 - 0.472iT - 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 5.70iT - 29T^{2} \)
31 \( 1 - 7.23iT - 31T^{2} \)
37 \( 1 - 5.23T + 37T^{2} \)
41 \( 1 + 6.47T + 41T^{2} \)
43 \( 1 + 12.9T + 43T^{2} \)
47 \( 1 - 2.47T + 47T^{2} \)
53 \( 1 - 8.47iT - 53T^{2} \)
59 \( 1 + 4.47T + 59T^{2} \)
61 \( 1 - 2.76iT - 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + 2.76iT - 71T^{2} \)
73 \( 1 + 6.76iT - 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 + 16.6T + 83T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 - 5.23iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.581970573275890258902804584812, −8.257727202885729411436940199168, −7.26969712866685221702542788923, −6.66498553078278138667165030181, −5.98279254890133807172330459390, −4.63762755889374355324945374256, −3.68074304791265306997459786622, −2.88987444622136810356448536851, −1.49463270926863549536525469824, −0.33422167430254205903067072716, 2.01706663257402181522387983089, 3.00524266907389298534696270949, 3.86167420635426188678849682444, 4.93131617687550189457514548608, 5.39246272762699435325855789867, 6.56106077076978569060464554775, 7.61708810754020303668367782385, 8.280250120189673961002885068455, 9.056576299346222354173028169924, 9.798878203925078526575674649705

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