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Properties

Label 2-168-168.107-c1-0-10
Degree $2$
Conductor $168$
Sign $0.531 + 0.846i$
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  + (−1.40 − 0.133i)2-s + (−1.71 + 0.207i)3-s + (1.96 + 0.375i)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 + (2.91 − 0.713i)9-s + (−1.13 + 1.59i)10-s + (3.82 − 2.20i)11-s + (−3.45 − 0.238i)12-s − 6.43i·13-s + (3.15 − 2.00i)14-s + (−0.941 + 2.20i)15-s + (3.71 + 1.47i)16-s + (2.52 − 1.45i)17-s + ⋯
L(s)  = 1  + (−0.995 − 0.0943i)2-s + (−0.992 + 0.119i)3-s + (0.982 + 0.187i)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 + (0.971 − 0.237i)9-s + (−0.358 + 0.504i)10-s + (1.15 − 0.665i)11-s + (−0.997 − 0.0689i)12-s − 1.78i·13-s + (0.843 − 0.536i)14-s + (−0.243 + 0.569i)15-s + (0.929 + 0.369i)16-s + (0.613 − 0.354i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.531 + 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.531 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.531 + 0.846i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.531 + 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.482427 - 0.266684i\)
\(L(\frac12)\) \(\approx\) \(0.482427 - 0.266684i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.133i)T \)
3 \( 1 + (1.71 - 0.207i)T \)
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

−12.43357492661038262208507220442, −11.52028611401446250862001083466, −10.57603136532822840884901658339, −9.561481910181245427972484940295, −8.882279749322634193129084933517, −7.41249763358598796288729480174, −6.18859600668768963972653150058, −5.44136723556407760464455234184, −3.22927083504299719977887685669, −0.876588589763659232315218558488, 1.62896232089681380008858815900, 3.95303320560584107435435913856, 5.96721949064859624247411089966, 6.77111424527588740858569427721, 7.39099696134007304650211081195, 9.359182227264380183458876713441, 9.805907333491307185226426126306, 10.90656175402802547186968899846, 11.70712874224979546265045209475, 12.57543821697058347878827632161

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