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Properties

Label 2-168-56.27-c3-0-38
Degree $2$
Conductor $168$
Sign $0.999 - 0.0330i$
Analytic cond. $9.91232$
Root an. cond. $3.14838$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.81 − 0.285i)2-s + 3i·3-s + (7.83 − 1.60i)4-s + 17.9·5-s + (0.857 + 8.44i)6-s + (4.95 − 17.8i)7-s + (21.5 − 6.77i)8-s − 9·9-s + (50.6 − 5.14i)10-s − 49.0·11-s + (4.82 + 23.5i)12-s − 3.98·13-s + (8.83 − 51.6i)14-s + 53.9i·15-s + (58.8 − 25.2i)16-s − 25.2i·17-s + ⋯
L(s)  = 1  + (0.994 − 0.101i)2-s + 0.577i·3-s + (0.979 − 0.201i)4-s + 1.60·5-s + (0.0583 + 0.574i)6-s + (0.267 − 0.963i)7-s + (0.954 − 0.299i)8-s − 0.333·9-s + (1.60 − 0.162i)10-s − 1.34·11-s + (0.116 + 0.565i)12-s − 0.0849·13-s + (0.168 − 0.985i)14-s + 0.928i·15-s + (0.919 − 0.394i)16-s − 0.360i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0330i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.999 - 0.0330i$
Analytic conductor: \(9.91232\)
Root analytic conductor: \(3.14838\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :3/2),\ 0.999 - 0.0330i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.85000 + 0.0636809i\)
\(L(\frac12)\) \(\approx\) \(3.85000 + 0.0636809i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.81 + 0.285i)T \)
3 \( 1 - 3iT \)
7 \( 1 + (-4.95 + 17.8i)T \)
good5 \( 1 - 17.9T + 125T^{2} \)
11 \( 1 + 49.0T + 1.33e3T^{2} \)
13 \( 1 + 3.98T + 2.19e3T^{2} \)
17 \( 1 + 25.2iT - 4.91e3T^{2} \)
19 \( 1 - 134. iT - 6.85e3T^{2} \)
23 \( 1 + 79.8iT - 1.21e4T^{2} \)
29 \( 1 - 210. iT - 2.43e4T^{2} \)
31 \( 1 + 107.T + 2.97e4T^{2} \)
37 \( 1 + 51.8iT - 5.06e4T^{2} \)
41 \( 1 - 414. iT - 6.89e4T^{2} \)
43 \( 1 + 186.T + 7.95e4T^{2} \)
47 \( 1 + 517.T + 1.03e5T^{2} \)
53 \( 1 + 442. iT - 1.48e5T^{2} \)
59 \( 1 + 225. iT - 2.05e5T^{2} \)
61 \( 1 - 111.T + 2.26e5T^{2} \)
67 \( 1 - 544.T + 3.00e5T^{2} \)
71 \( 1 - 942. iT - 3.57e5T^{2} \)
73 \( 1 + 96.0iT - 3.89e5T^{2} \)
79 \( 1 + 1.22e3iT - 4.93e5T^{2} \)
83 \( 1 + 365. iT - 5.71e5T^{2} \)
89 \( 1 - 395. iT - 7.04e5T^{2} \)
97 \( 1 + 1.07e3iT - 9.12e5T^{2} \)
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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

−12.75797517893249933721988712961, −11.17792836836624102335053728721, −10.27686757523772778890580003942, −9.910299742093184813161999274725, −8.077163209309565019518951198298, −6.70172779433412838342784034547, −5.55341704270376018795492816726, −4.80619012462533202447675096375, −3.22312334494018403576443058536, −1.78477662094404150854377140717, 1.98216784628198458567260955589, 2.72046078874176814447864130153, 5.10129454751239279586661557270, 5.67188460633542311363042269565, 6.68884293966003895048719733767, 7.987541841735820794342659006365, 9.307201284423389349139727853642, 10.53537758122929104618353957053, 11.55034474530674533926804054759, 12.72361224677691400538109286399

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