Properties

Label 2-168-24.11-c1-0-11
Degree $2$
Conductor $168$
Sign $0.816 + 0.577i$
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  − 1.41·2-s + (1 − 1.41i)3-s + 2.00·4-s + 2.82·5-s + (−1.41 + 2.00i)6-s + i·7-s − 2.82·8-s + (−1.00 − 2.82i)9-s − 4.00·10-s + 5.65i·11-s + (2.00 − 2.82i)12-s − 4i·13-s − 1.41i·14-s + (2.82 − 4.00i)15-s + 4.00·16-s − 2.82i·17-s + ⋯
L(s)  = 1  − 1.00·2-s + (0.577 − 0.816i)3-s + 1.00·4-s + 1.26·5-s + (−0.577 + 0.816i)6-s + 0.377i·7-s − 1.00·8-s + (−0.333 − 0.942i)9-s − 1.26·10-s + 1.70i·11-s + (0.577 − 0.816i)12-s − 1.10i·13-s − 0.377i·14-s + (0.730 − 1.03i)15-s + 1.00·16-s − 0.685i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.816 + 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00317 - 0.318845i\)
\(L(\frac12)\) \(\approx\) \(1.00317 - 0.318845i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 + (-1 + 1.41i)T \)
7 \( 1 - iT \)
good5 \( 1 - 2.82T + 5T^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + 5.65T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + 2.82iT - 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 5.65T + 53T^{2} \)
59 \( 1 - 8.48iT - 59T^{2} \)
61 \( 1 - 12iT - 61T^{2} \)
67 \( 1 - 6T + 67T^{2} \)
71 \( 1 + 8.48T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 8iT - 79T^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 - 2.82iT - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−12.71302018579111683343399594800, −11.79409364499894445803609984470, −10.25728989822941167181456325337, −9.610473966394151717831103315116, −8.719073890077931542499335182637, −7.53128917723646651206989951814, −6.68528013093385233077056108291, −5.49382774102635178437791917358, −2.74296786811854716823813494487, −1.72582014390493601132796804977, 2.01262040497359072834218006410, 3.55523307825622633029163165586, 5.54540533642596049058520222448, 6.58899892600173745310000664358, 8.170281262010503247767184607189, 9.016118934977986001684017364111, 9.688542091161208903672529595085, 10.71806530856034042165341477219, 11.28685791098586714203294707302, 13.11572623526138878803455348210

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