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Properties

Label 2-168-168.125-c1-0-7
Degree $2$
Conductor $168$
Sign $0.970 - 0.242i$
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  − 1.41i·2-s + (0.420 + 1.68i)3-s − 2.00·4-s + 2.16i·5-s + (2.37 − 0.595i)6-s + 2.64·7-s + 2.82i·8-s + (−2.64 + 1.41i)9-s + 3.06·10-s + (−0.841 − 3.36i)12-s + 5.59·13-s − 3.74i·14-s + (−3.64 + 0.913i)15-s + 4.00·16-s + (2 + 3.74i)18-s − 8.66·19-s + ⋯
L(s)  = 1  − 0.999i·2-s + (0.242 + 0.970i)3-s − 1.00·4-s + 0.970i·5-s + (0.970 − 0.242i)6-s + 0.999·7-s + 1.00i·8-s + (−0.881 + 0.471i)9-s + 0.970·10-s + (−0.242 − 0.970i)12-s + 1.55·13-s − 1.00i·14-s + (−0.941 + 0.235i)15-s + 1.00·16-s + (0.471 + 0.881i)18-s − 1.98·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17865 + 0.145376i\)
\(L(\frac12)\) \(\approx\) \(1.17865 + 0.145376i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (-0.420 - 1.68i)T \)
7 \( 1 - 2.64T \)
good5 \( 1 - 2.16iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 5.59T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 8.66T + 19T^{2} \)
23 \( 1 + 7.48iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 14.4iT - 59T^{2} \)
61 \( 1 - 0.543T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 5.29T + 79T^{2} \)
83 \( 1 + 1.40iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−12.75967009231250941928500506151, −11.30219314043004826845158339324, −10.85849251062633611087059298779, −10.27796110072031520011874203767, −8.773866269396189631506373058201, −8.276092541079059357045226719953, −6.22892797271015656627944433071, −4.69480786319982795740479484200, −3.72715325844416900736938918670, −2.33356906938141667109848102754, 1.37254558373175719219593004940, 4.05588669246356563725010332985, 5.41948169760209339774001169767, 6.40843084066942294344561401782, 7.71401341340265144867451957575, 8.514182484684504403762543275500, 8.999845339415762303950435506685, 10.86630621472003022872340978413, 12.08904995614324607837772177847, 13.13453508804746263165965123240

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