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Properties

Label 2-168-168.5-c1-0-18
Degree $2$
Conductor $168$
Sign $0.938 - 0.344i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.37 + 0.323i)2-s + (1.70 + 0.283i)3-s + (1.79 + 0.890i)4-s + (−2.24 − 1.29i)5-s + (2.26 + 0.942i)6-s + (−2.53 + 0.751i)7-s + (2.17 + 1.80i)8-s + (2.83 + 0.967i)9-s + (−2.67 − 2.51i)10-s + (−1.63 − 2.83i)11-s + (2.80 + 2.02i)12-s − 0.912·13-s + (−3.73 + 0.214i)14-s + (−3.47 − 2.85i)15-s + (2.41 + 3.18i)16-s + (−2.39 − 4.14i)17-s + ⋯
L(s)  = 1  + (0.973 + 0.228i)2-s + (0.986 + 0.163i)3-s + (0.895 + 0.445i)4-s + (−1.00 − 0.580i)5-s + (0.923 + 0.384i)6-s + (−0.958 + 0.284i)7-s + (0.769 + 0.638i)8-s + (0.946 + 0.322i)9-s + (−0.846 − 0.795i)10-s + (−0.493 − 0.855i)11-s + (0.810 + 0.585i)12-s − 0.253·13-s + (−0.998 + 0.0572i)14-s + (−0.897 − 0.737i)15-s + (0.603 + 0.797i)16-s + (−0.580 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.938 - 0.344i$
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: \((2,\ 168,\ (\ :1/2),\ 0.938 - 0.344i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07254 + 0.368604i\)
\(L(\frac12)\) \(\approx\) \(2.07254 + 0.368604i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 - 0.323i)T \)
3 \( 1 + (-1.70 - 0.283i)T \)
7 \( 1 + (2.53 - 0.751i)T \)
good5 \( 1 + (2.24 + 1.29i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.63 + 2.83i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.912T + 13T^{2} \)
17 \( 1 + (2.39 + 4.14i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.66 - 4.61i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.45 - 2.57i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.35T + 29T^{2} \)
31 \( 1 + (-8.18 + 4.72i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.59 + 0.922i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.91T + 41T^{2} \)
43 \( 1 - 8.00iT - 43T^{2} \)
47 \( 1 + (3.29 - 5.70i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.841 + 1.45i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.50 - 0.867i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.72 + 8.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.8 + 6.27i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.603iT - 71T^{2} \)
73 \( 1 + (1.29 - 0.746i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.0625 - 0.108i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.246iT - 83T^{2} \)
89 \( 1 + (1.80 - 3.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.10iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.02942361959689160822440521063, −12.18924802115817814178699425732, −11.13465596048661886890042302585, −9.732384762863786199044973063857, −8.470549289626051456316929189479, −7.74201929700137100107931727553, −6.49367416808772968124210309629, −4.94148459187719850545050245285, −3.77747040583234867593579540976, −2.78546641453389050173910249535, 2.51949199455668292325114454047, 3.57545533958334949801436332504, 4.60270480301807287903239710301, 6.76050830263650684736096538775, 7.11537990602668074670061234614, 8.508487341264054406385679683358, 10.01202072621643167564366805595, 10.75349313253224810605950436262, 12.08437106641805831966717552527, 12.90921025009565521742279592640

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