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Properties

Label 2-168-168.11-c1-0-15
Degree $2$
Conductor $168$
Sign $0.934 + 0.356i$
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  + (0.819 − 1.15i)2-s + (1.03 + 1.38i)3-s + (−0.656 − 1.88i)4-s + (0.692 + 1.19i)5-s + (2.44 − 0.0625i)6-s + (2.08 + 1.62i)7-s + (−2.71 − 0.791i)8-s + (−0.839 + 2.88i)9-s + (1.94 + 0.184i)10-s + (−3.82 − 2.20i)11-s + (1.93 − 2.87i)12-s − 6.43i·13-s + (3.58 − 1.07i)14-s + (−0.941 + 2.20i)15-s + (−3.13 + 2.48i)16-s + (−2.52 − 1.45i)17-s + ⋯
L(s)  = 1  + (0.579 − 0.814i)2-s + (0.600 + 0.799i)3-s + (−0.328 − 0.944i)4-s + (0.309 + 0.536i)5-s + (0.999 − 0.0255i)6-s + (0.789 + 0.613i)7-s + (−0.960 − 0.279i)8-s + (−0.279 + 0.960i)9-s + (0.616 + 0.0584i)10-s + (−1.15 − 0.665i)11-s + (0.558 − 0.829i)12-s − 1.78i·13-s + (0.957 − 0.287i)14-s + (−0.243 + 0.569i)15-s + (−0.784 + 0.620i)16-s + (−0.613 − 0.354i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.934 + 0.356i$
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: \((2,\ 168,\ (\ :1/2),\ 0.934 + 0.356i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72761 - 0.318100i\)
\(L(\frac12)\) \(\approx\) \(1.72761 - 0.318100i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.819 + 1.15i)T \)
3 \( 1 + (-1.03 - 1.38i)T \)
7 \( 1 + (-2.08 - 1.62i)T \)
good5 \( 1 + (-0.692 - 1.19i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.82 + 2.20i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.43iT - 13T^{2} \)
17 \( 1 + (2.52 + 1.45i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.58 - 2.75i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.84 - 3.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.67T + 29T^{2} \)
31 \( 1 + (-2.17 - 1.25i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.00 - 2.88i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.497iT - 41T^{2} \)
43 \( 1 - 0.865T + 43T^{2} \)
47 \( 1 + (-1.59 - 2.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.12 + 7.15i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.62 - 3.82i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.99 + 1.72i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.36 - 5.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.90T + 71T^{2} \)
73 \( 1 + (-3.23 + 5.60i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.65 + 0.953i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.00iT - 83T^{2} \)
89 \( 1 + (-8.22 + 4.75i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.19T + 97T^{2} \)
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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

−12.93347033278821770536279831392, −11.51961367118350984386069213677, −10.65800531536094275337236117426, −10.12977427651316714365213228866, −8.836956016452668811536841769521, −7.85957531636010036724587735675, −5.66184384398357837923769035754, −5.07557816564391199650956324322, −3.34894706046395306238535621943, −2.45776682557778957994718715084, 2.15575844107689154384793024393, 4.14204244743904919230990113874, 5.21625462379649671697834150294, 6.79776713102920551275777684169, 7.43365618459036817202302502156, 8.545446668852573607565445294740, 9.330857403114970215463095814198, 11.16515418187029125935942099258, 12.27076368564128719179593734067, 13.20412555492286271405957234752

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