Properties

Label 2-1680-5.4-c1-0-31
Degree $2$
Conductor $1680$
Sign $-0.447 + 0.894i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + (1 − 2i)5-s + i·7-s − 9-s − 2·11-s − 6i·13-s + (2 + i)15-s + 4i·17-s − 6·19-s − 21-s − 8i·23-s + (−3 − 4i)25-s i·27-s − 6·29-s + 2·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.447 − 0.894i)5-s + 0.377i·7-s − 0.333·9-s − 0.603·11-s − 1.66i·13-s + (0.516 + 0.258i)15-s + 0.970i·17-s − 1.37·19-s − 0.218·21-s − 1.66i·23-s + (−0.600 − 0.800i)25-s − 0.192i·27-s − 1.11·29-s + 0.359·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8901756374\)
\(L(\frac12)\) \(\approx\) \(0.8901756374\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + (-1 + 2i)T \)
7 \( 1 - iT \)
good11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + 8iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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

−8.956316066605758678104660264142, −8.396241310581547500698757024770, −7.895034858739627332379328569961, −6.36813478017749726723996899578, −5.76530168964035672627355246907, −4.98026452858904035159920529753, −4.22826466880037989105653819923, −3.01882639645572028465256492280, −1.98790849289310377535660918738, −0.31175079890528843742727667392, 1.67735342354335653842118722384, 2.47408740578303534096102298417, 3.60143476294605514230152633665, 4.64020074084346486807634327314, 5.79966116986227678363823251171, 6.46024747802954457681897307897, 7.32109692112122257685867019217, 7.63725665974598548433790479494, 9.034433095617039822921498366773, 9.448757242994790628898179101176

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