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Properties

Label 2-168-168.5-c1-0-27
Degree $2$
Conductor $168$
Sign $-0.963 + 0.267i$
Analytic cond. $1.34148$
Root an. cond. $1.15822$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.368 − 1.36i)2-s + (−0.390 − 1.68i)3-s + (−1.72 − 1.00i)4-s + (−1.54 − 0.894i)5-s + (−2.44 − 0.0881i)6-s + (2.63 + 0.230i)7-s + (−2.00 + 1.99i)8-s + (−2.69 + 1.31i)9-s + (−1.79 + 1.78i)10-s + (−0.501 − 0.868i)11-s + (−1.02 + 3.30i)12-s + 2.47·13-s + (1.28 − 3.51i)14-s + (−0.904 + 2.96i)15-s + (1.97 + 3.47i)16-s + (−3.32 − 5.76i)17-s + ⋯
L(s)  = 1  + (0.260 − 0.965i)2-s + (−0.225 − 0.974i)3-s + (−0.864 − 0.502i)4-s + (−0.692 − 0.399i)5-s + (−0.999 − 0.0360i)6-s + (0.996 + 0.0869i)7-s + (−0.710 + 0.703i)8-s + (−0.898 + 0.439i)9-s + (−0.566 + 0.564i)10-s + (−0.151 − 0.261i)11-s + (−0.294 + 0.955i)12-s + 0.685·13-s + (0.343 − 0.939i)14-s + (−0.233 + 0.765i)15-s + (0.494 + 0.869i)16-s + (−0.807 − 1.39i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.963 + 0.267i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ -0.963 + 0.267i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.135608 - 0.996589i\)
\(L(\frac12)\) \(\approx\) \(0.135608 - 0.996589i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.368 + 1.36i)T \)
3 \( 1 + (0.390 + 1.68i)T \)
7 \( 1 + (-2.63 - 0.230i)T \)
good5 \( 1 + (1.54 + 0.894i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.501 + 0.868i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.47T + 13T^{2} \)
17 \( 1 + (3.32 + 5.76i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.85 + 3.22i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.85 - 3.95i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.748T + 29T^{2} \)
31 \( 1 + (-2.87 + 1.65i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.22 + 1.86i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.01T + 41T^{2} \)
43 \( 1 - 9.19iT - 43T^{2} \)
47 \( 1 + (-1.19 + 2.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.33 - 10.9i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.34 + 4.24i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.02 - 3.50i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.89 - 3.98i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.46iT - 71T^{2} \)
73 \( 1 + (5.68 - 3.28i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.53 + 4.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.65iT - 83T^{2} \)
89 \( 1 + (7.39 - 12.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.75iT - 97T^{2} \)
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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

−12.12765270737388711893673070424, −11.41297455015094321076035174754, −11.01063519784469213749236459157, −9.150719267028347491690667182661, −8.352868773140906353116216061724, −7.21624687863394684756204975046, −5.53424934966677352465296504347, −4.54096878309429985263825548021, −2.74038272444029503990431712233, −1.00447522015302056029986445267, 3.59299770508658165538807470125, 4.51023570115278900626199459260, 5.62707729053860891988601596777, 6.91759098290872004039071015442, 8.202427590713049593523126202036, 8.831155582053932557081189339894, 10.35468236987309787457546864025, 11.16516559393985439739512081997, 12.22197372357444516136270884782, 13.51795270321520604089076738893

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