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Properties

Label 2-168-7.5-c2-0-1
Degree $2$
Conductor $168$
Sign $0.211 - 0.977i$
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 + (7.79 + 4.49i)5-s + (−6.97 − 0.602i)7-s + (1.5 − 2.59i)9-s + (2.31 + 4.01i)11-s + 15.7i·13-s − 15.5·15-s + (−8.01 + 4.62i)17-s + (23.8 + 13.7i)19-s + (10.9 − 5.13i)21-s + (−2.82 + 4.89i)23-s + (27.9 + 48.4i)25-s + 5.19i·27-s − 14.6·29-s + (8.63 − 4.98i)31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (1.55 + 0.899i)5-s + (−0.996 − 0.0860i)7-s + (0.166 − 0.288i)9-s + (0.210 + 0.365i)11-s + 1.21i·13-s − 1.03·15-s + (−0.471 + 0.272i)17-s + (1.25 + 0.724i)19-s + (0.522 − 0.244i)21-s + (−0.122 + 0.212i)23-s + (1.11 + 1.93i)25-s + 0.192i·27-s − 0.506·29-s + (0.278 − 0.160i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.211 - 0.977i$
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: \((2,\ 168,\ (\ :1),\ 0.211 - 0.977i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.09490 + 0.883483i\)
\(L(\frac12)\) \(\approx\) \(1.09490 + 0.883483i\)
\(L(2)\) 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 \( 1 + (6.97 + 0.602i)T \)
good5 \( 1 + (-7.79 - 4.49i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-2.31 - 4.01i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 15.7iT - 169T^{2} \)
17 \( 1 + (8.01 - 4.62i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-23.8 - 13.7i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (2.82 - 4.89i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 14.6T + 841T^{2} \)
31 \( 1 + (-8.63 + 4.98i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-25.4 + 44.0i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 70.2iT - 1.68e3T^{2} \)
43 \( 1 + 49.9T + 1.84e3T^{2} \)
47 \( 1 + (14.3 + 8.26i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-4.28 - 7.41i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-85.0 + 49.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (30 + 17.3i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (5.19 + 8.99i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 93.3T + 5.04e3T^{2} \)
73 \( 1 + (-107. + 62.2i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (55.3 - 95.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 136. iT - 6.88e3T^{2} \)
89 \( 1 + (138. + 80.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 129. iT - 9.40e3T^{2} \)
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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

−12.87168746951867298807546358203, −11.68349149411113486222271197150, −10.59259923818205335401092055405, −9.731580023222418529050975030233, −9.281542302969531159755848998172, −7.09322610998539102955775909119, −6.38102292089287650700225034641, −5.46583719329394114341274435583, −3.67309562755208395564197507001, −2.04646568196334993523976557555, 0.970453822212403832767519869345, 2.82430819470137303186868442035, 5.01436583029365228578744391573, 5.84099587734797068774216603839, 6.73778737485978947046686267010, 8.378588575719016604764936103844, 9.560892934715234846594970546184, 10.03690904434744367396930361266, 11.43327926324807263611546680553, 12.65232614339331842144840713974

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