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Properties

Label 2-168-168.101-c1-0-15
Degree $2$
Conductor $168$
Sign $-0.741 + 0.670i$
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  + (−1.30 − 0.550i)2-s + (−1.09 + 1.34i)3-s + (1.39 + 1.43i)4-s + (−2.46 + 1.42i)5-s + (2.16 − 1.14i)6-s + (−1.02 − 2.44i)7-s + (−1.02 − 2.63i)8-s + (−0.614 − 2.93i)9-s + (4.00 − 0.497i)10-s + (2.42 − 4.20i)11-s + (−3.45 + 0.306i)12-s − 2.75·13-s + (−0.0151 + 3.74i)14-s + (0.780 − 4.87i)15-s + (−0.115 + 3.99i)16-s + (−1.75 + 3.03i)17-s + ⋯
L(s)  = 1  + (−0.921 − 0.389i)2-s + (−0.630 + 0.776i)3-s + (0.696 + 0.717i)4-s + (−1.10 + 0.637i)5-s + (0.883 − 0.469i)6-s + (−0.385 − 0.922i)7-s + (−0.362 − 0.931i)8-s + (−0.204 − 0.978i)9-s + (1.26 − 0.157i)10-s + (0.731 − 1.26i)11-s + (−0.996 + 0.0885i)12-s − 0.763·13-s + (−0.00403 + 0.999i)14-s + (0.201 − 1.25i)15-s + (−0.0289 + 0.999i)16-s + (−0.425 + 0.736i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 + 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.741 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.741 + 0.670i$
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: \((2,\ 168,\ (\ :1/2),\ -0.741 + 0.670i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0503756 - 0.130861i\)
\(L(\frac12)\) \(\approx\) \(0.0503756 - 0.130861i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 + 0.550i)T \)
3 \( 1 + (1.09 - 1.34i)T \)
7 \( 1 + (1.02 + 2.44i)T \)
good5 \( 1 + (2.46 - 1.42i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.42 + 4.20i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.75T + 13T^{2} \)
17 \( 1 + (1.75 - 3.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.14 + 5.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.15 - 1.82i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.90T + 29T^{2} \)
31 \( 1 + (0.858 + 0.495i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.06 - 0.614i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.10T + 41T^{2} \)
43 \( 1 - 5.11iT - 43T^{2} \)
47 \( 1 + (5.61 + 9.72i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.00 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.890 - 0.514i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.24 + 2.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.02 + 2.90i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.75iT - 71T^{2} \)
73 \( 1 + (-0.291 - 0.168i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.80 + 4.85i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.138iT - 83T^{2} \)
89 \( 1 + (0.580 + 1.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 11.0iT - 97T^{2} \)
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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

−11.82339556538886302946996535006, −11.19191267121860245210639037063, −10.62712873410113536978698125867, −9.561275932360647181306067562561, −8.461620325343867043436195759274, −7.19977638249358024525724196320, −6.34452725427943651969788916147, −4.13717784136787700412206056394, −3.34707661340753047193363404768, −0.18075267526204347612488644068, 2.02114874750242464587435590579, 4.67411643611529768557453379763, 6.00722226898397061786952587651, 7.10803314217665950902011065292, 7.919609800948433375914374998750, 8.950360691099214455022499385153, 10.01731567665028435368332490644, 11.41639347881169673148432400624, 12.19484234023065977318068503940, 12.54127630432932671249674910758

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