Properties

Degree $2$
Conductor $168$
Sign $0.605 + 0.795i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + (−1 − 1.73i)5-s + (2.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (3 − 5.19i)11-s − 3·13-s − 1.99·15-s + (−2 + 3.46i)17-s + (2.5 + 4.33i)19-s + (2 − 1.73i)21-s + (2 + 3.46i)23-s + (0.500 − 0.866i)25-s − 0.999·27-s − 4·29-s + (−3.5 + 6.06i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.447 − 0.774i)5-s + (0.944 + 0.327i)7-s + (−0.166 − 0.288i)9-s + (0.904 − 1.56i)11-s − 0.832·13-s − 0.516·15-s + (−0.485 + 0.840i)17-s + (0.573 + 0.993i)19-s + (0.436 − 0.377i)21-s + (0.417 + 0.722i)23-s + (0.100 − 0.173i)25-s − 0.192·27-s − 0.742·29-s + (−0.628 + 1.08i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.605 + 0.795i$
Motivic weight: \(1\)
Character: $\chi_{168} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11967 - 0.555054i\)
\(L(\frac12)\) \(\approx\) \(1.11967 - 0.555054i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2.5 - 0.866i)T \)
good5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3T + 13T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.5 - 7.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4 - 6.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.5 + 12.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-4 - 6.92i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−12.49647059581523935845655071106, −11.80810672278820619933441987182, −10.93984127357960550447846175547, −9.267857396078675292804220194486, −8.455168944411838828307103830164, −7.74939842826450915271496415580, −6.22004389258745725172411486743, −4.99206537175416333524925458485, −3.51752340962738967894359338796, −1.45509825848733654655920391659, 2.39332483157576102990205156124, 4.10700387459099160093599550272, 4.98644188984549650667574953911, 7.06202613067146614143981319815, 7.46661102487355113936002959526, 9.066908169917119266659675428788, 9.865810713128648151345970312920, 11.10923750331065684197624462419, 11.63213957703065132972287836513, 12.93998168163479237060150960111

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