Properties

Label 2-1680-420.419-c0-0-13
Degree $2$
Conductor $1680$
Sign $-0.866 + 0.5i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)3-s + i·5-s i·7-s + (0.499 + 0.866i)9-s − 11-s − 1.73·13-s + (0.5 − 0.866i)15-s i·17-s + (−0.5 + 0.866i)21-s − 25-s − 0.999i·27-s − 1.73i·29-s + (0.866 + 0.5i)33-s + 35-s + (1.49 + 0.866i)39-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + i·5-s i·7-s + (0.499 + 0.866i)9-s − 11-s − 1.73·13-s + (0.5 − 0.866i)15-s i·17-s + (−0.5 + 0.866i)21-s − 25-s − 0.999i·27-s − 1.73i·29-s + (0.866 + 0.5i)33-s + 35-s + (1.49 + 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.866 + 0.5i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (1679, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :0),\ -0.866 + 0.5i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2759450500\)
\(L(\frac12)\) \(\approx\) \(0.2759450500\)
\(L(1)\) 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 \( 1 - iT \)
7 \( 1 + iT \)
good11 \( 1 + T + T^{2} \)
13 \( 1 + 1.73T + T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 1.73T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 2T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 1.73iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - 1.73T + T^{2} \)
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

−9.630525234398433106608012478768, −7.87197206788914487850500623746, −7.54527316966163752238153546058, −6.93760253299501339743973329889, −6.11841742206581296925811140549, −5.08690416411274267394770612853, −4.43459636065572891161287494863, −2.99762567472172961268955978983, −2.12410374074404881861011431649, −0.22042260962690694920908119877, 1.74813273874283228670279689593, 3.03815347176741101706990309041, 4.38980477155746988146585901402, 5.17160460920898847488586842526, 5.44274076881664188763265179329, 6.49508175963059169814886375249, 7.54615149348236272974007763268, 8.425302477408888190511985331131, 9.172422217338236728612709185603, 9.887337019820227846483613637431

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