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Properties

Label 2-168-168.11-c1-0-22
Degree $2$
Conductor $168$
Sign $-0.822 + 0.569i$
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  + (−1.18 − 0.765i)2-s + (0.728 − 1.57i)3-s + (0.826 + 1.82i)4-s + (−1.00 − 1.73i)5-s + (−2.06 + 1.31i)6-s + (−2.50 − 0.837i)7-s + (0.412 − 2.79i)8-s + (−1.93 − 2.28i)9-s + (−0.137 + 2.82i)10-s + (3.33 + 1.92i)11-s + (3.46 + 0.0269i)12-s − 2.05i·13-s + (2.34 + 2.91i)14-s + (−3.45 + 0.310i)15-s + (−2.63 + 3.01i)16-s + (−4.92 − 2.84i)17-s + ⋯
L(s)  = 1  + (−0.840 − 0.541i)2-s + (0.420 − 0.907i)3-s + (0.413 + 0.910i)4-s + (−0.447 − 0.775i)5-s + (−0.844 + 0.535i)6-s + (−0.948 − 0.316i)7-s + (0.145 − 0.989i)8-s + (−0.646 − 0.762i)9-s + (−0.0436 + 0.894i)10-s + (1.00 + 0.580i)11-s + (0.999 + 0.00778i)12-s − 0.571i·13-s + (0.626 + 0.779i)14-s + (−0.891 + 0.0802i)15-s + (−0.658 + 0.752i)16-s + (−1.19 − 0.689i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 + 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.822 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.822 + 0.569i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ -0.822 + 0.569i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.208385 - 0.667274i\)
\(L(\frac12)\) \(\approx\) \(0.208385 - 0.667274i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.18 + 0.765i)T \)
3 \( 1 + (-0.728 + 1.57i)T \)
7 \( 1 + (2.50 + 0.837i)T \)
good5 \( 1 + (1.00 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.33 - 1.92i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.05iT - 13T^{2} \)
17 \( 1 + (4.92 + 2.84i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.232 + 0.403i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.711 - 1.23i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 9.25T + 29T^{2} \)
31 \( 1 + (-4.99 - 2.88i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.569 - 0.328i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.42iT - 41T^{2} \)
43 \( 1 - 6.77T + 43T^{2} \)
47 \( 1 + (-1.79 - 3.10i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.50 + 2.60i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.72 - 3.30i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.50 + 4.91i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.78 - 6.55i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 + (2.02 - 3.50i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.00488 + 0.00281i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.56iT - 83T^{2} \)
89 \( 1 + (8.14 - 4.69i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.17T + 97T^{2} \)
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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

−12.33632070945795294191941698853, −11.67118819632540616621373001668, −10.22964692201965288213037419603, −9.107472002715548627018833107729, −8.538959507772041959153699745206, −7.26005903335095532226332192190, −6.54614478129518956025686930952, −4.17794185691416630918352772749, −2.72038328481193142075709226892, −0.840113678566120014304151721430, 2.78188120751031025384751189471, 4.25996899546503906839505810501, 6.11322934725463532586024894789, 6.85458325717814458544060149201, 8.391394096656544385074628108106, 9.067764049677489638986369329076, 10.02752364102240399151025673649, 10.91817744124219815639270884095, 11.75730078047517721004650835632, 13.57372256932328682786324321358

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