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Properties

Label 2-168-56.19-c1-0-0
Degree $2$
Conductor $168$
Sign $-0.710 + 0.703i$
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  + (−0.647 + 1.25i)2-s + (−0.866 + 0.5i)3-s + (−1.16 − 1.62i)4-s + (−1.61 + 2.79i)5-s + (−0.0679 − 1.41i)6-s + (−1.82 − 1.91i)7-s + (2.79 − 0.406i)8-s + (0.499 − 0.866i)9-s + (−2.46 − 3.83i)10-s + (−1.10 − 1.91i)11-s + (1.82 + 0.829i)12-s − 5.08·13-s + (3.58 − 1.05i)14-s − 3.22i·15-s + (−1.30 + 3.78i)16-s + (−2.73 + 1.57i)17-s + ⋯
L(s)  = 1  + (−0.457 + 0.889i)2-s + (−0.499 + 0.288i)3-s + (−0.580 − 0.814i)4-s + (−0.721 + 1.25i)5-s + (−0.0277 − 0.576i)6-s + (−0.690 − 0.723i)7-s + (0.989 − 0.143i)8-s + (0.166 − 0.288i)9-s + (−0.781 − 1.21i)10-s + (−0.333 − 0.577i)11-s + (0.525 + 0.239i)12-s − 1.40·13-s + (0.959 − 0.282i)14-s − 0.833i·15-s + (−0.325 + 0.945i)16-s + (−0.663 + 0.383i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.710 + 0.703i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ -0.710 + 0.703i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0674163 - 0.164036i\)
\(L(\frac12)\) \(\approx\) \(0.0674163 - 0.164036i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.647 - 1.25i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (1.82 + 1.91i)T \)
good5 \( 1 + (1.61 - 2.79i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.10 + 1.91i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.08T + 13T^{2} \)
17 \( 1 + (2.73 - 1.57i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.93 - 1.69i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.65 + 1.53i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 9.88iT - 29T^{2} \)
31 \( 1 + (-1.01 - 1.75i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.798 - 0.460i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.96iT - 41T^{2} \)
43 \( 1 + 6.68T + 43T^{2} \)
47 \( 1 + (1.06 - 1.83i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.12 + 1.80i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (10.6 - 6.14i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.34 + 10.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.40 + 7.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.7iT - 71T^{2} \)
73 \( 1 + (-7.82 + 4.51i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-9.60 - 5.54i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.57iT - 83T^{2} \)
89 \( 1 + (6.32 + 3.65i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.2iT - 97T^{2} \)
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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

−13.72866526629709247035921698389, −12.38367730848684206372546712873, −10.99980318785460368935636150413, −10.42304421158874699033947657544, −9.530817539702648579736234917358, −7.963456334270321345878028279925, −7.06641860084028020749780039585, −6.37117697040220676116569109971, −4.86316136166993630654289263605, −3.40044032444536143800080525607, 0.19571935003522894014867638410, 2.39568271217269483058336740129, 4.31387757325786885068354276053, 5.28913235121921851760337275786, 7.21882117561627295119158426426, 8.199605445242853256541596588754, 9.331715983305968459441890667555, 9.993444684750665672258967126996, 11.65602171136463956046497101373, 12.00162776047921555648192389505

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