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Properties

Label 2-168-168.107-c1-0-0
Degree $2$
Conductor $168$
Sign $-0.393 - 0.919i$
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  + (−1.32 − 0.480i)2-s + (−0.316 − 1.70i)3-s + (1.53 + 1.27i)4-s + (−1.86 + 3.22i)5-s + (−0.397 + 2.41i)6-s + (−2.46 − 0.962i)7-s + (−1.43 − 2.44i)8-s + (−2.79 + 1.07i)9-s + (4.02 − 3.39i)10-s + (−2.26 + 1.31i)11-s + (1.69 − 3.02i)12-s + 3.57i·13-s + (2.81 + 2.46i)14-s + (6.07 + 2.14i)15-s + (0.729 + 3.93i)16-s + (0.186 − 0.107i)17-s + ⋯
L(s)  = 1  + (−0.940 − 0.339i)2-s + (−0.182 − 0.983i)3-s + (0.768 + 0.639i)4-s + (−0.832 + 1.44i)5-s + (−0.162 + 0.986i)6-s + (−0.931 − 0.363i)7-s + (−0.505 − 0.862i)8-s + (−0.933 + 0.359i)9-s + (1.27 − 1.07i)10-s + (−0.684 + 0.395i)11-s + (0.488 − 0.872i)12-s + 0.990i·13-s + (0.752 + 0.658i)14-s + (1.56 + 0.554i)15-s + (0.182 + 0.983i)16-s + (0.0451 − 0.0260i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.393 - 0.919i$
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: \((2,\ 168,\ (\ :1/2),\ -0.393 - 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.105762 + 0.160399i\)
\(L(\frac12)\) \(\approx\) \(0.105762 + 0.160399i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 + 0.480i)T \)
3 \( 1 + (0.316 + 1.70i)T \)
7 \( 1 + (2.46 + 0.962i)T \)
good5 \( 1 + (1.86 - 3.22i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.26 - 1.31i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.57iT - 13T^{2} \)
17 \( 1 + (-0.186 + 0.107i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.14 - 1.97i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.33 + 4.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.57T + 29T^{2} \)
31 \( 1 + (4.26 - 2.46i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.31 + 4.22i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.909iT - 41T^{2} \)
43 \( 1 - 3.73T + 43T^{2} \)
47 \( 1 + (-0.586 + 1.01i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.06 - 1.83i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.79 - 3.92i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.301 - 0.173i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.98 + 8.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 + (-4.45 - 7.71i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.70 - 5.60i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.73iT - 83T^{2} \)
89 \( 1 + (-15.4 - 8.93i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.88T + 97T^{2} \)
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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

−12.72539427646272286834934342384, −12.02131517779405690476268161391, −10.93846038625686623257911134750, −10.43919598940872457354071355407, −9.002502134901830045434750813770, −7.64685046380791109223534596377, −7.08687357950759094983457931467, −6.34970170638846564113117545854, −3.57954664686268914428477652317, −2.38671658647423532142383804052, 0.23062545579855576336669478542, 3.29057538252528440483170931365, 5.03270351821405601749247903769, 5.84737127565929793058598661622, 7.62722864277541408542748901084, 8.661243168949474366401592467873, 9.233393348858892911742994025963, 10.27762038518581624112613722385, 11.28340488488244359183917971559, 12.27984494971258602961153321764

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