↑

Properties

Label 2-168-168.107-c1-0-22
Degree $2$
Conductor $168$
Sign $0.979 + 0.201i$
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  + (1.40 + 0.133i)2-s + (1.03 − 1.38i)3-s + (1.96 + 0.375i)4-s + (−0.692 + 1.19i)5-s + (1.64 − 1.81i)6-s + (−2.08 + 1.62i)7-s + (2.71 + 0.791i)8-s + (−0.839 − 2.88i)9-s + (−1.13 + 1.59i)10-s + (−3.82 + 2.20i)11-s + (2.56 − 2.33i)12-s − 6.43i·13-s + (−3.15 + 2.00i)14-s + (0.941 + 2.20i)15-s + (3.71 + 1.47i)16-s + (−2.52 + 1.45i)17-s + ⋯
L(s)  = 1  + (0.995 + 0.0943i)2-s + (0.600 − 0.799i)3-s + (0.982 + 0.187i)4-s + (−0.309 + 0.536i)5-s + (0.672 − 0.739i)6-s + (−0.789 + 0.613i)7-s + (0.960 + 0.279i)8-s + (−0.279 − 0.960i)9-s + (−0.358 + 0.504i)10-s + (−1.15 + 0.665i)11-s + (0.739 − 0.672i)12-s − 1.78i·13-s + (−0.843 + 0.536i)14-s + (0.243 + 0.569i)15-s + (0.929 + 0.369i)16-s + (−0.613 + 0.354i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.979 + 0.201i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.979 + 0.201i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05212 - 0.208877i\)
\(L(\frac12)\) \(\approx\) \(2.05212 - 0.208877i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.133i)T \)
3 \( 1 + (-1.03 + 1.38i)T \)
7 \( 1 + (2.08 - 1.62i)T \)
good5 \( 1 + (0.692 - 1.19i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.82 - 2.20i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.43iT - 13T^{2} \)
17 \( 1 + (2.52 - 1.45i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.58 + 2.75i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.84 - 3.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.67T + 29T^{2} \)
31 \( 1 + (2.17 - 1.25i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.00 - 2.88i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.497iT - 41T^{2} \)
43 \( 1 - 0.865T + 43T^{2} \)
47 \( 1 + (1.59 - 2.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.12 + 7.15i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.62 + 3.82i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.99 + 1.72i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.36 + 5.83i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.90T + 71T^{2} \)
73 \( 1 + (-3.23 - 5.60i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.65 + 0.953i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.00iT - 83T^{2} \)
89 \( 1 + (-8.22 - 4.75i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.19T + 97T^{2} \)
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

−12.94542147375776104418988063894, −12.23623381130520465618895371530, −11.00882670447874285913289513388, −9.889617177674507467150742904263, −8.228267470133032458370487326428, −7.41498456002291547978801374192, −6.41221993619698572154685828889, −5.25962541969046223343766985431, −3.29421648544679727924078174522, −2.57577872430886430191442501320, 2.64395545975944627537156500039, 3.99363868236579170423691693450, 4.75628245842784715182053556810, 6.25040461412674261131656102942, 7.57295638823530370417673069973, 8.799509390489343368914397451869, 10.01478027517234157869975973715, 10.85479871123460391500616368288, 11.93403230833559656678372307804, 13.08669367218865396584331138014

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