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Properties

Label 2-168-168.5-c1-0-5
Degree $2$
Conductor $168$
Sign $-0.276 - 0.960i$
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  + (−0.998 + 1.00i)2-s + (0.390 + 1.68i)3-s + (−0.00659 − 1.99i)4-s + (1.54 + 0.894i)5-s + (−2.08 − 1.29i)6-s + (2.63 + 0.230i)7-s + (2.00 + 1.99i)8-s + (−2.69 + 1.31i)9-s + (−2.44 + 0.658i)10-s + (0.501 + 0.868i)11-s + (3.37 − 0.792i)12-s − 2.47·13-s + (−2.86 + 2.41i)14-s + (−0.904 + 2.96i)15-s + (−3.99 + 0.0263i)16-s + (−3.32 − 5.76i)17-s + ⋯
L(s)  = 1  + (−0.705 + 0.708i)2-s + (0.225 + 0.974i)3-s + (−0.00329 − 0.999i)4-s + (0.692 + 0.399i)5-s + (−0.849 − 0.528i)6-s + (0.996 + 0.0869i)7-s + (0.710 + 0.703i)8-s + (−0.898 + 0.439i)9-s + (−0.772 + 0.208i)10-s + (0.151 + 0.261i)11-s + (0.973 − 0.228i)12-s − 0.685·13-s + (−0.764 + 0.644i)14-s + (−0.233 + 0.765i)15-s + (−0.999 + 0.00659i)16-s + (−0.807 − 1.39i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.276 - 0.960i$
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: \((2,\ 168,\ (\ :1/2),\ -0.276 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.597743 + 0.794155i\)
\(L(\frac12)\) \(\approx\) \(0.597743 + 0.794155i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.998 - 1.00i)T \)
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

−13.64043823237657268490785298657, −11.62882145215073466512596254091, −10.80519573696726124032244652748, −9.844629036107436029080113886853, −9.163875623546569544570273483708, −8.080652925622285221583954032222, −6.90680291842493838146868094879, −5.48057274465507297595271789518, −4.64021867845192934728007096028, −2.32726957215975913717354084240, 1.37570155427610002387879104964, 2.58789442981405495727976601237, 4.62441754964436348600338499754, 6.36104563841320582265555774112, 7.57423283467712565948956298195, 8.561268008739580609431112448696, 9.206951885531377592236420494069, 10.70374857725386098805850008449, 11.40527829911559607031112008201, 12.62982350559623117745019201084

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