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Properties

Label 2-168-7.3-c2-0-3
Degree $2$
Conductor $168$
Sign $0.670 - 0.742i$
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 + (−2.05 + 1.18i)5-s + (4.55 + 5.31i)7-s + (1.5 + 2.59i)9-s + (4.00 − 6.94i)11-s + 23.8i·13-s + 4.10·15-s + (27.9 + 16.1i)17-s + (12.6 − 7.28i)19-s + (−2.24 − 11.9i)21-s + (−2.82 − 4.89i)23-s + (−9.69 + 16.7i)25-s − 5.19i·27-s − 34.3·29-s + (23.5 + 13.5i)31-s + ⋯
L(s)  = 1  + (−0.5 − 0.288i)3-s + (−0.410 + 0.236i)5-s + (0.651 + 0.758i)7-s + (0.166 + 0.288i)9-s + (0.364 − 0.631i)11-s + 1.83i·13-s + 0.273·15-s + (1.64 + 0.950i)17-s + (0.663 − 0.383i)19-s + (−0.106 − 0.567i)21-s + (−0.122 − 0.212i)23-s + (−0.387 + 0.671i)25-s − 0.192i·27-s − 1.18·29-s + (0.759 + 0.438i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 - 0.742i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.670 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.670 - 0.742i$
Analytic conductor: \(4.57766\)
Root analytic conductor: \(2.13954\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1),\ 0.670 - 0.742i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.12145 + 0.498245i\)
\(L(\frac12)\) \(\approx\) \(1.12145 + 0.498245i\)
\(L(2)\) 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 \( 1 + (-4.55 - 5.31i)T \)
good5 \( 1 + (2.05 - 1.18i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-4.00 + 6.94i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 23.8iT - 169T^{2} \)
17 \( 1 + (-27.9 - 16.1i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-12.6 + 7.28i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (2.82 + 4.89i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 34.3T + 841T^{2} \)
31 \( 1 + (-23.5 - 13.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-4.04 - 7.00i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 50.6iT - 1.68e3T^{2} \)
43 \( 1 + 39.8T + 1.84e3T^{2} \)
47 \( 1 + (8.13 - 4.69i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-27.0 + 46.8i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (38.1 + 22.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (30 - 17.3i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-58.9 + 102. i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 38.2T + 5.04e3T^{2} \)
73 \( 1 + (-50.4 - 29.1i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-38.2 - 66.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 58.2iT - 6.88e3T^{2} \)
89 \( 1 + (-12.5 + 7.21i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 120. iT - 9.40e3T^{2} \)
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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

−12.34263888870486779831539259335, −11.69597662555914207016454455157, −11.07522073282043743059888521181, −9.628898546147554853130214466833, −8.534928664421657496554892191206, −7.45860063805434406490813243393, −6.27056505190834632358113553753, −5.18731248961981873135267366773, −3.67673331108487302815544080180, −1.66953012511966268488505732258, 0.910307767994797916885318421456, 3.43531865926732640806766932416, 4.76213127693962594941926462258, 5.74192828205467348433522470137, 7.46876349966275498410577080788, 7.982120567969383001439939339307, 9.729984245456010029351639726833, 10.33843619279663489772486572578, 11.55141751629780543282484312127, 12.20655160862950994139512814982

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