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Properties

Label 2-168-168.125-c3-0-67
Degree $2$
Conductor $168$
Sign $-0.389 + 0.920i$
Analytic cond. $9.91232$
Root an. cond. $3.14838$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.24 + 1.72i)2-s + (1.04 + 5.08i)3-s + (2.04 − 7.73i)4-s − 3.71i·5-s + (−11.1 − 9.59i)6-s + (15.6 − 9.97i)7-s + (8.78 + 20.8i)8-s + (−24.8 + 10.6i)9-s + (6.40 + 8.31i)10-s − 48.1·11-s + (41.5 + 2.27i)12-s − 69.5·13-s + (−17.7 + 49.2i)14-s + (18.8 − 3.89i)15-s + (−55.6 − 31.5i)16-s − 48.9·17-s + ⋯
L(s)  = 1  + (−0.792 + 0.610i)2-s + (0.201 + 0.979i)3-s + (0.255 − 0.966i)4-s − 0.332i·5-s + (−0.757 − 0.652i)6-s + (0.842 − 0.538i)7-s + (0.388 + 0.921i)8-s + (−0.918 + 0.395i)9-s + (0.202 + 0.263i)10-s − 1.31·11-s + (0.998 + 0.0546i)12-s − 1.48·13-s + (−0.338 + 0.940i)14-s + (0.325 − 0.0670i)15-s + (−0.869 − 0.493i)16-s − 0.697·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.389 + 0.920i$
Analytic conductor: \(9.91232\)
Root analytic conductor: \(3.14838\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :3/2),\ -0.389 + 0.920i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0247890 - 0.0374147i\)
\(L(\frac12)\) \(\approx\) \(0.0247890 - 0.0374147i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.24 - 1.72i)T \)
3 \( 1 + (-1.04 - 5.08i)T \)
7 \( 1 + (-15.6 + 9.97i)T \)
good5 \( 1 + 3.71iT - 125T^{2} \)
11 \( 1 + 48.1T + 1.33e3T^{2} \)
13 \( 1 + 69.5T + 2.19e3T^{2} \)
17 \( 1 + 48.9T + 4.91e3T^{2} \)
19 \( 1 + 62.2T + 6.85e3T^{2} \)
23 \( 1 + 63.4iT - 1.21e4T^{2} \)
29 \( 1 - 20.2T + 2.43e4T^{2} \)
31 \( 1 - 77.3iT - 2.97e4T^{2} \)
37 \( 1 - 151. iT - 5.06e4T^{2} \)
41 \( 1 + 284.T + 6.89e4T^{2} \)
43 \( 1 + 441. iT - 7.95e4T^{2} \)
47 \( 1 + 615.T + 1.03e5T^{2} \)
53 \( 1 - 129.T + 1.48e5T^{2} \)
59 \( 1 - 406. iT - 2.05e5T^{2} \)
61 \( 1 - 576.T + 2.26e5T^{2} \)
67 \( 1 - 665. iT - 3.00e5T^{2} \)
71 \( 1 + 129. iT - 3.57e5T^{2} \)
73 \( 1 - 290. iT - 3.89e5T^{2} \)
79 \( 1 + 130.T + 4.93e5T^{2} \)
83 \( 1 + 763. iT - 5.71e5T^{2} \)
89 \( 1 - 269.T + 7.04e5T^{2} \)
97 \( 1 + 1.37e3iT - 9.12e5T^{2} \)
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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

−11.63347682155594164264196601749, −10.48659868926033934694672054612, −10.16547589977507235563097377317, −8.776617777943789183486325946026, −8.137876860277663915837452100977, −6.98337219649159407748436649527, −5.19063625301567308162710770735, −4.68889575335567080442104267801, −2.37005421435984611201042725652, −0.02379348270174148272291994593, 1.98717401680132640996237198723, 2.80493188585368539262322982454, 4.99339290380930648158078738457, 6.72312645780052523704946493953, 7.75107185212278168997123216970, 8.365095806264125442383752262258, 9.573845869823800315465549176860, 10.80885610323475760343308582252, 11.57967358267832207144601421805, 12.56212789588671738960249288790

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