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Properties

Label 2-168-168.107-c1-0-16
Degree $2$
Conductor $168$
Sign $0.838 - 0.544i$
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  + (−0.389 + 1.35i)2-s + (1.72 − 0.143i)3-s + (−1.69 − 1.05i)4-s + (1.77 − 3.07i)5-s + (−0.476 + 2.40i)6-s + (−0.793 + 2.52i)7-s + (2.09 − 1.89i)8-s + (2.95 − 0.496i)9-s + (3.49 + 3.61i)10-s + (0.396 − 0.229i)11-s + (−3.08 − 1.58i)12-s − 0.799i·13-s + (−3.12 − 2.06i)14-s + (2.62 − 5.57i)15-s + (1.75 + 3.59i)16-s + (−5.48 + 3.16i)17-s + ⋯
L(s)  = 1  + (−0.275 + 0.961i)2-s + (0.996 − 0.0829i)3-s + (−0.848 − 0.529i)4-s + (0.795 − 1.37i)5-s + (−0.194 + 0.980i)6-s + (−0.299 + 0.953i)7-s + (0.742 − 0.670i)8-s + (0.986 − 0.165i)9-s + (1.10 + 1.14i)10-s + (0.119 − 0.0690i)11-s + (−0.889 − 0.456i)12-s − 0.221i·13-s + (−0.834 − 0.550i)14-s + (0.678 − 1.43i)15-s + (0.439 + 0.898i)16-s + (−1.33 + 0.767i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.838 - 0.544i$
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: \((2,\ 168,\ (\ :1/2),\ 0.838 - 0.544i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29610 + 0.384122i\)
\(L(\frac12)\) \(\approx\) \(1.29610 + 0.384122i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.389 - 1.35i)T \)
3 \( 1 + (-1.72 + 0.143i)T \)
7 \( 1 + (0.793 - 2.52i)T \)
good5 \( 1 + (-1.77 + 3.07i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.396 + 0.229i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.799iT - 13T^{2} \)
17 \( 1 + (5.48 - 3.16i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.61 - 4.53i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.55 + 2.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.60T + 29T^{2} \)
31 \( 1 + (4.42 - 2.55i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.89 - 1.67i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 - 3.56T + 43T^{2} \)
47 \( 1 + (2.15 - 3.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.16 - 2.02i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.49 + 0.864i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.60 + 2.66i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.00979 - 0.0169i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.04T + 71T^{2} \)
73 \( 1 + (-4.15 - 7.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.8 - 6.28i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.694iT - 83T^{2} \)
89 \( 1 + (7.02 + 4.05i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.05T + 97T^{2} \)
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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

−12.84530246476976521684661615087, −12.68173813138232664392652514333, −10.38420355206782137837843212768, −9.204986424475591669424513156881, −8.886864707183135871616497147266, −8.073493860637842233273141642481, −6.49575245828188247740094723172, −5.47809644650966945747586255945, −4.20443228483992500885797833568, −1.86923160073967519579101624938, 2.16609593909952094185110894725, 3.18733309352038886421432989424, 4.43289653265045351907557347860, 6.72295003717251523640886803235, 7.52454480173692302594736229977, 9.096868576061271843069791207927, 9.695172078425514049974316975852, 10.68432455633231926383174035485, 11.27596167371403372940779689151, 13.20552204411596049068405390037

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