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Properties

Label 2-168-7.5-c2-0-0
Degree $2$
Conductor $168$
Sign $0.422 - 0.906i$
Analytic cond. $4.57766$
Root an. cond. $2.13954$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)3-s + (−2.31 − 1.33i)5-s + (6.66 + 2.13i)7-s + (1.5 − 2.59i)9-s + (5.85 + 10.1i)11-s + 20.4i·13-s + 4.63·15-s + (−14.2 + 8.21i)17-s + (26.9 + 15.5i)19-s + (−11.8 + 2.57i)21-s + (2.82 − 4.89i)23-s + (−8.91 − 15.4i)25-s + 5.19i·27-s + 21.6·29-s + (−4.60 + 2.65i)31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (−0.463 − 0.267i)5-s + (0.952 + 0.304i)7-s + (0.166 − 0.288i)9-s + (0.531 + 0.921i)11-s + 1.57i·13-s + 0.308·15-s + (−0.837 + 0.483i)17-s + (1.41 + 0.818i)19-s + (−0.564 + 0.122i)21-s + (0.122 − 0.212i)23-s + (−0.356 − 0.617i)25-s + 0.192i·27-s + 0.746·29-s + (−0.148 + 0.0857i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.422 - 0.906i$
Analytic conductor: \(4.57766\)
Root analytic conductor: \(2.13954\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1),\ 0.422 - 0.906i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.02834 + 0.655377i\)
\(L(\frac12)\) \(\approx\) \(1.02834 + 0.655377i\)
\(L(2)\) 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 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 + (-6.66 - 2.13i)T \)
good5 \( 1 + (2.31 + 1.33i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-5.85 - 10.1i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 20.4iT - 169T^{2} \)
17 \( 1 + (14.2 - 8.21i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-26.9 - 15.5i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-2.82 + 4.89i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 21.6T + 841T^{2} \)
31 \( 1 + (4.60 - 2.65i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (9.64 - 16.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 28.0iT - 1.68e3T^{2} \)
43 \( 1 - 73.0T + 1.84e3T^{2} \)
47 \( 1 + (27.9 + 16.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (47.3 + 82.0i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (97.8 - 56.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (30 + 17.3i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-3.84 - 6.65i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 22.9T + 5.04e3T^{2} \)
73 \( 1 + (5.83 - 3.37i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-23.5 + 40.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 39.2iT - 6.88e3T^{2} \)
89 \( 1 + (-87.5 - 50.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 30.0iT - 9.40e3T^{2} \)
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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

−12.24018336203538829102337046871, −11.89027436333818256378886221530, −10.94805548161677867778112131894, −9.677829459546847999505489347455, −8.740652491437481838235265580611, −7.52090711808703030836097174866, −6.35287379940609543932633655435, −4.86642043747093376909499313203, −4.10996778585643199135857649549, −1.73233168854476431102271607833, 0.887586230714801006567079825071, 3.13608882854240365148848919335, 4.76098316946319380751964944855, 5.86721238964887469673297242504, 7.28790785716971607623447259276, 7.998747817428375022771866100050, 9.303463320279947567830246463712, 10.88933880441071401892830462468, 11.18637138379464793289978314007, 12.21067047516377159856232789586

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