Properties

Label 2-168-168.101-c1-0-19
Degree $2$
Conductor $168$
Sign $-0.156 + 0.987i$
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 + (−0.609 − 1.62i)3-s + (1.79 − 0.890i)4-s + (2.24 − 1.29i)5-s + (1.36 + 2.03i)6-s + (−2.53 − 0.751i)7-s + (−2.17 + 1.80i)8-s + (−2.25 + 1.97i)9-s + (−2.67 + 2.51i)10-s + (1.63 − 2.83i)11-s + (−2.53 − 2.36i)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 + (−0.351 − 0.936i)3-s + (0.895 − 0.445i)4-s + (1.00 − 0.580i)5-s + (0.556 + 0.830i)6-s + (−0.958 − 0.284i)7-s + (−0.769 + 0.638i)8-s + (−0.752 + 0.658i)9-s + (−0.846 + 0.795i)10-s + (0.493 − 0.855i)11-s + (−0.731 − 0.681i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ -0.156 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.434948 - 0.509182i\)
\(L(\frac12)\) \(\approx\) \(0.434948 - 0.509182i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.609 + 1.62i)T \)
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

−12.42394819853691461209662471696, −11.53064342762322074533502688194, −10.31601049526535425806677455007, −9.388840534899183721347308801516, −8.519884775868176765358707960997, −7.18039184713783458086471500986, −6.33044094209337259361617931185, −5.46715240226268151958948483198, −2.63158985340794623632906315417, −0.901018907686549829448010697352, 2.36999276275939176518137570099, 3.87330493430253502357882815815, 6.01681944085831980345523838894, 6.47746665041379507586122484321, 8.219977154330479023945498706822, 9.546254272515194197228004678356, 9.968391474403148109069910029663, 10.55268058443672084819440437227, 11.95257396703685472915741720300, 12.65554536240801029243480219948

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