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Properties

Label 2-168-168.107-c1-0-15
Degree $2$
Conductor $168$
Sign $0.895 + 0.444i$
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  + (−1.40 − 0.191i)2-s + (1.66 + 0.493i)3-s + (1.92 + 0.535i)4-s + (1.09 − 1.89i)5-s + (−2.23 − 1.00i)6-s + (−0.451 − 2.60i)7-s + (−2.59 − 1.11i)8-s + (2.51 + 1.63i)9-s + (−1.89 + 2.44i)10-s + (−1.45 + 0.837i)11-s + (2.93 + 1.84i)12-s + 1.56i·13-s + (0.134 + 3.73i)14-s + (2.74 − 2.60i)15-s + (3.42 + 2.06i)16-s + (−0.278 + 0.160i)17-s + ⋯
L(s)  = 1  + (−0.990 − 0.135i)2-s + (0.958 + 0.285i)3-s + (0.963 + 0.267i)4-s + (0.488 − 0.845i)5-s + (−0.911 − 0.412i)6-s + (−0.170 − 0.985i)7-s + (−0.918 − 0.395i)8-s + (0.837 + 0.546i)9-s + (−0.598 + 0.772i)10-s + (−0.437 + 0.252i)11-s + (0.847 + 0.531i)12-s + 0.432i·13-s + (0.0360 + 0.999i)14-s + (0.709 − 0.671i)15-s + (0.856 + 0.516i)16-s + (−0.0676 + 0.0390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.895 + 0.444i$
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: \((2,\ 168,\ (\ :1/2),\ 0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03986 - 0.243923i\)
\(L(\frac12)\) \(\approx\) \(1.03986 - 0.243923i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.191i)T \)
3 \( 1 + (-1.66 - 0.493i)T \)
7 \( 1 + (0.451 + 2.60i)T \)
good5 \( 1 + (-1.09 + 1.89i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.45 - 0.837i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.56iT - 13T^{2} \)
17 \( 1 + (0.278 - 0.160i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.87 + 4.97i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.26 - 5.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.04T + 29T^{2} \)
31 \( 1 + (7.76 - 4.48i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.946 - 0.546i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.44iT - 41T^{2} \)
43 \( 1 + 11.6T + 43T^{2} \)
47 \( 1 + (2.25 - 3.90i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.42 + 7.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.37 + 3.10i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.80 - 2.19i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.716 - 1.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.37T + 71T^{2} \)
73 \( 1 + (4.49 + 7.77i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.58 + 2.07i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.06iT - 83T^{2} \)
89 \( 1 + (-4.57 - 2.64i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.23T + 97T^{2} \)
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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

−12.92180982768892066912189461932, −11.50268732798604875868145908659, −10.29087142620368347271005467148, −9.611122899419814644809220417785, −8.824070439160240220228404010050, −7.77165894805617437671425469963, −6.85610521708067659416526357753, −4.91465296100028961726091296011, −3.30964842663194320103253453849, −1.59522635659105395110869557634, 2.15233386550754000059974311409, 3.11712107354151336769277963077, 5.79118253206917112785127236324, 6.75181232843014596426745392505, 7.957293794478883098143463024899, 8.679809299593629668195851636660, 9.838352787553420748733133234882, 10.41104323168853364742458663864, 11.83414525131187620937622445338, 12.78383731979940718722213965819

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