Properties

Label 2-168-168.5-c1-0-8
Degree $2$
Conductor $168$
Sign $-0.154 - 0.988i$
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  + (1.07 + 0.917i)2-s + (−0.0574 + 1.73i)3-s + (0.317 + 1.97i)4-s + (0.461 + 0.266i)5-s + (−1.64 + 1.81i)6-s + (0.489 − 2.60i)7-s + (−1.46 + 2.41i)8-s + (−2.99 − 0.198i)9-s + (0.252 + 0.710i)10-s + (−2.28 − 3.96i)11-s + (−3.43 + 0.435i)12-s + 4.97·13-s + (2.91 − 2.35i)14-s + (−0.487 + 0.783i)15-s + (−3.79 + 1.25i)16-s + (2.16 + 3.74i)17-s + ⋯
L(s)  = 1  + (0.761 + 0.648i)2-s + (−0.0331 + 0.999i)3-s + (0.158 + 0.987i)4-s + (0.206 + 0.119i)5-s + (−0.673 + 0.739i)6-s + (0.184 − 0.982i)7-s + (−0.519 + 0.854i)8-s + (−0.997 − 0.0663i)9-s + (0.0798 + 0.224i)10-s + (−0.689 − 1.19i)11-s + (−0.992 + 0.125i)12-s + 1.38·13-s + (0.778 − 0.628i)14-s + (−0.125 + 0.202i)15-s + (−0.949 + 0.313i)16-s + (0.524 + 0.907i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.154 - 0.988i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ -0.154 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07664 + 1.25770i\)
\(L(\frac12)\) \(\approx\) \(1.07664 + 1.25770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.07 - 0.917i)T \)
3 \( 1 + (0.0574 - 1.73i)T \)
7 \( 1 + (-0.489 + 2.60i)T \)
good5 \( 1 + (-0.461 - 0.266i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.28 + 3.96i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.97T + 13T^{2} \)
17 \( 1 + (-2.16 - 3.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.921 - 1.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.103 + 0.0596i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.74T + 29T^{2} \)
31 \( 1 + (1.93 - 1.11i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.02 + 4.05i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.60T + 41T^{2} \)
43 \( 1 - 1.87iT - 43T^{2} \)
47 \( 1 + (-2.91 + 5.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.29 - 3.97i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.71 - 3.29i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.07 - 1.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.4 - 6.05i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.20iT - 71T^{2} \)
73 \( 1 + (-8.35 + 4.82i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.0228 - 0.0396i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.86iT - 83T^{2} \)
89 \( 1 + (8.23 - 14.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.18iT - 97T^{2} \)
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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

−13.62893009069681358704542395965, −12.13962141148738200145995752897, −10.85888865933908091922407801962, −10.44805519656785970648875612779, −8.667189834511790552293927383610, −8.037362741598641283048095552914, −6.36367044823122065970557569331, −5.54100643170657471559029363189, −4.14939917035060803925685780030, −3.29736191654829264380649068222, 1.71407151699329298752400282239, 2.97028377650107347997880534057, 4.96812585601088954868736165919, 5.88595628478146647846752194026, 7.02943276274255629211876095094, 8.451731252455355338231100529604, 9.586528964309799800252825392667, 10.89110380570261216381292345787, 11.85875857686233172318954170528, 12.48858267911812478132648830213

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