↑

Properties

Label 2-168-8.5-c1-0-2
Degree $2$
Conductor $168$
Sign $-0.204 - 0.978i$
Analytic cond. $1.34148$
Root an. cond. $1.15822$
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  + (−0.621 + 1.27i)2-s − i·3-s + (−1.22 − 1.57i)4-s + 3.69i·5-s + (1.27 + 0.621i)6-s + 7-s + (2.76 − 0.578i)8-s − 9-s + (−4.69 − 2.29i)10-s + 3.21i·11-s + (−1.57 + 1.22i)12-s + 5.08i·13-s + (−0.621 + 1.27i)14-s + 3.69·15-s + (−0.985 + 3.87i)16-s + 0.616·17-s + ⋯
L(s)  = 1  + (−0.439 + 0.898i)2-s − 0.577i·3-s + (−0.613 − 0.789i)4-s + 1.65i·5-s + (0.518 + 0.253i)6-s + 0.377·7-s + (0.978 − 0.204i)8-s − 0.333·9-s + (−1.48 − 0.726i)10-s + 0.968i·11-s + (−0.455 + 0.354i)12-s + 1.40i·13-s + (−0.166 + 0.339i)14-s + 0.954·15-s + (−0.246 + 0.969i)16-s + 0.149·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.204 - 0.978i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ -0.204 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.548411 + 0.674911i\)
\(L(\frac12)\) \(\approx\) \(0.548411 + 0.674911i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.621 - 1.27i)T \)
3 \( 1 + iT \)
7 \( 1 - T \)
good5 \( 1 - 3.69iT - 5T^{2} \)
11 \( 1 - 3.21iT - 11T^{2} \)
13 \( 1 - 5.08iT - 13T^{2} \)
17 \( 1 - 0.616T + 17T^{2} \)
19 \( 1 + 4.48iT - 19T^{2} \)
23 \( 1 - 1.38T + 23T^{2} \)
29 \( 1 + 5.67iT - 29T^{2} \)
31 \( 1 - 6.91T + 31T^{2} \)
37 \( 1 + 6.91iT - 37T^{2} \)
41 \( 1 + 0.616T + 41T^{2} \)
43 \( 1 + 7.99iT - 43T^{2} \)
47 \( 1 - 4.97T + 47T^{2} \)
53 \( 1 - 4.48iT - 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 - 12.4iT - 61T^{2} \)
67 \( 1 + 9.56iT - 67T^{2} \)
71 \( 1 + 15.2T + 71T^{2} \)
73 \( 1 - 15.5T + 73T^{2} \)
79 \( 1 + 5.23T + 79T^{2} \)
83 \( 1 + 10.4iT - 83T^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 - 9.73T + 97T^{2} \)
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

−13.56595288691378825552415699376, −11.92321505634523931496990731544, −10.97090856263877634946752539661, −9.990831585468625805558084144893, −8.891427433256147819506526007248, −7.41976815603693061426088283119, −7.03725074680728373261064844951, −6.07928040900194048345284184862, −4.39319609981274216690734538171, −2.24266150120173149050534248796, 1.05711256563279271078940209044, 3.28629990387395524987637807875, 4.64899190421121670788093227993, 5.55124937070516634115880602087, 8.165006013063358051711600204769, 8.386189466857837991939941909299, 9.554949806983700459724878774191, 10.44795355393710915680963044605, 11.50347128808890876403063018237, 12.44691001257220343385101869948

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