↑

Properties

Label 2-168-168.5-c1-0-14
Degree $2$
Conductor $168$
Sign $0.693 - 0.720i$
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  + (0.717 + 1.21i)2-s + (1.11 − 1.32i)3-s + (−0.971 + 1.74i)4-s + (0.337 + 0.195i)5-s + (2.41 + 0.401i)6-s + (1.39 + 2.24i)7-s + (−2.82 + 0.0704i)8-s + (−0.529 − 2.95i)9-s + (0.00458 + 0.551i)10-s + (−0.748 − 1.29i)11-s + (1.24 + 3.23i)12-s + 3.28·13-s + (−1.73 + 3.31i)14-s + (0.634 − 0.232i)15-s + (−2.11 − 3.39i)16-s + (−1.68 − 2.91i)17-s + ⋯
L(s)  = 1  + (0.507 + 0.861i)2-s + (0.641 − 0.766i)3-s + (−0.485 + 0.874i)4-s + (0.151 + 0.0872i)5-s + (0.986 + 0.164i)6-s + (0.527 + 0.849i)7-s + (−0.999 + 0.0249i)8-s + (−0.176 − 0.984i)9-s + (0.00144 + 0.174i)10-s + (−0.225 − 0.390i)11-s + (0.358 + 0.933i)12-s + 0.911·13-s + (−0.464 + 0.885i)14-s + (0.163 − 0.0599i)15-s + (−0.528 − 0.848i)16-s + (−0.407 − 0.706i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.693 - 0.720i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.693 - 0.720i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55418 + 0.660921i\)
\(L(\frac12)\) \(\approx\) \(1.55418 + 0.660921i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.717 - 1.21i)T \)
3 \( 1 + (-1.11 + 1.32i)T \)
7 \( 1 + (-1.39 - 2.24i)T \)
good5 \( 1 + (-0.337 - 0.195i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.748 + 1.29i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.28T + 13T^{2} \)
17 \( 1 + (1.68 + 2.91i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.56 - 4.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.72 + 2.72i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.13T + 29T^{2} \)
31 \( 1 + (3.60 - 2.07i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.46 - 4.31i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 + 4.79iT - 43T^{2} \)
47 \( 1 + (2.51 - 4.34i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.499 + 0.864i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.36 + 0.785i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.40 + 5.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.05 + 1.76i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.3iT - 71T^{2} \)
73 \( 1 + (-2.76 + 1.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.239 - 0.414i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 17.4iT - 83T^{2} \)
89 \( 1 + (-2.54 + 4.41i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.00iT - 97T^{2} \)
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

−13.06184779044745487196500673411, −12.30755213175570381997785737226, −11.29284068373055175806778196402, −9.427914396327494359059316020086, −8.411852968014419067716508023808, −7.88756967181435679821969597360, −6.44671677562867397671723077230, −5.71353656411123395638906381090, −3.99175898905261210996230515162, −2.43401401642573580471252798073, 2.01204065205952192747816406954, 3.73653766108167827150187749057, 4.48311518166199324488276551382, 5.83690754241104142686159390298, 7.68779094598316587303878462038, 8.929749721887275897312341524218, 9.799869400436738809930094962868, 10.87267836071858866350301473627, 11.26734029236166023118629178524, 13.05544679212223144781047519927

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