↑

Properties

Label 2-168-56.27-c1-0-2
Degree $2$
Conductor $168$
Sign $-0.651 - 0.758i$
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  + (0.733 + 1.20i)2-s + i·3-s + (−0.924 + 1.77i)4-s − 1.12·5-s + (−1.20 + 0.733i)6-s + (−2.11 + 1.59i)7-s + (−2.82 + 0.181i)8-s − 9-s + (−0.826 − 1.36i)10-s + 5.11·11-s + (−1.77 − 0.924i)12-s + 5.88·13-s + (−3.47 − 1.38i)14-s − 1.12i·15-s + (−2.28 − 3.28i)16-s − 3.31i·17-s + ⋯
L(s)  = 1  + (0.518 + 0.855i)2-s + 0.577i·3-s + (−0.462 + 0.886i)4-s − 0.504·5-s + (−0.493 + 0.299i)6-s + (−0.798 + 0.601i)7-s + (−0.997 + 0.0641i)8-s − 0.333·9-s + (−0.261 − 0.431i)10-s + 1.54·11-s + (−0.511 − 0.267i)12-s + 1.63·13-s + (−0.928 − 0.371i)14-s − 0.291i·15-s + (−0.572 − 0.820i)16-s − 0.804i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.651 - 0.758i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ -0.651 - 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.525974 + 1.14528i\)
\(L(\frac12)\) \(\approx\) \(0.525974 + 1.14528i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.733 - 1.20i)T \)
3 \( 1 - iT \)
7 \( 1 + (2.11 - 1.59i)T \)
good5 \( 1 + 1.12T + 5T^{2} \)
11 \( 1 - 5.11T + 11T^{2} \)
13 \( 1 - 5.88T + 13T^{2} \)
17 \( 1 + 3.31iT - 17T^{2} \)
19 \( 1 - 7.49iT - 19T^{2} \)
23 \( 1 + 1.73iT - 23T^{2} \)
29 \( 1 + 5.88iT - 29T^{2} \)
31 \( 1 - 6.04T + 31T^{2} \)
37 \( 1 - 1.65iT - 37T^{2} \)
41 \( 1 + 1.45iT - 41T^{2} \)
43 \( 1 - 1.79T + 43T^{2} \)
47 \( 1 + 5.56T + 47T^{2} \)
53 \( 1 + 3.62iT - 53T^{2} \)
59 \( 1 + 0.767iT - 59T^{2} \)
61 \( 1 - 0.317T + 61T^{2} \)
67 \( 1 + 6.56T + 67T^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 + 6.63iT - 73T^{2} \)
79 \( 1 + 3.01iT - 79T^{2} \)
83 \( 1 + 16.0iT - 83T^{2} \)
89 \( 1 - 8.08iT - 89T^{2} \)
97 \( 1 + 0.357iT - 97T^{2} \)
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

−13.37795199389568511373366600090, −12.06777352181974034734144310911, −11.60511377965772297473491505208, −9.869514416960885495878501288896, −8.928010651791116866662067708716, −8.071195601905205778378691537091, −6.48525634715945028509319285099, −5.88468310230999011218109607310, −4.17288677036512002759694033689, −3.43566717239621721667127941098, 1.19762532828351337275529244831, 3.32503399541183936896278137793, 4.21159672078248039943685690187, 6.10349341769786237560683828621, 6.83944041163827527604915229393, 8.572279089344954521088418682272, 9.454462978210158029675894910563, 10.83518254955454077029469075701, 11.47824488224394112962636111405, 12.45148535129843470724033481712

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