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Properties

Label 2-168-168.125-c1-0-15
Degree $2$
Conductor $168$
Sign $0.925 - 0.377i$
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 + 1.73i·3-s + 2.00·4-s − 3.46i·5-s + 2.44i·6-s + (1 + 2.44i)7-s + 2.82·8-s − 2.99·9-s − 4.89i·10-s − 5.65·11-s + 3.46i·12-s + (1.41 + 3.46i)14-s + 5.99·15-s + 4.00·16-s − 4.24·18-s + ⋯
L(s)  = 1  + 1.00·2-s + 0.999i·3-s + 1.00·4-s − 1.54i·5-s + 0.999i·6-s + (0.377 + 0.925i)7-s + 1.00·8-s − 0.999·9-s − 1.54i·10-s − 1.70·11-s + 1.00i·12-s + (0.377 + 0.925i)14-s + 1.54·15-s + 1.00·16-s − 0.999·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.925 - 0.377i$
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: \((2,\ 168,\ (\ :1/2),\ 0.925 - 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89180 + 0.371289i\)
\(L(\frac12)\) \(\approx\) \(1.89180 + 0.371289i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
7 \( 1 + (-1 - 2.44i)T \)
good5 \( 1 + 3.46iT - 5T^{2} \)
11 \( 1 + 5.65T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 + 4.89iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 14.1T + 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 9.79iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 17.3iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 19.5iT - 97T^{2} \)
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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

−12.86131953738232336514662291424, −12.02604047946620102182383686540, −11.08036469120415910863863826436, −9.937081285027590154317039486039, −8.731907194950073710484927293545, −7.88548266371998143322691787302, −5.61829327842256544630642368453, −5.26155228062265800101000909279, −4.23884153285095206597732003327, −2.51057144729191734177692515939, 2.30097137970135329042072321867, 3.39483135222866222223506198993, 5.23929776029237819618289289337, 6.48828852458800911734825677052, 7.32251673755100357297580495237, 7.915212305677035958632268223823, 10.43028162570420874848942155522, 10.81759116083119454036084346830, 11.80473006998180449808861758932, 13.03963336061986994798562147697

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