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Properties

Label 2-168-56.37-c1-0-15
Degree $2$
Conductor $168$
Sign $-0.999 - 0.0298i$
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  + (0.491 − 1.32i)2-s + (−0.866 + 0.5i)3-s + (−1.51 − 1.30i)4-s + (−3.08 − 1.78i)5-s + (0.236 + 1.39i)6-s + (−2.38 + 1.14i)7-s + (−2.47 + 1.36i)8-s + (0.499 − 0.866i)9-s + (−3.88 + 3.21i)10-s + (3.52 − 2.03i)11-s + (1.96 + 0.371i)12-s − 1.44i·13-s + (0.350 + 3.72i)14-s + 3.56·15-s + (0.595 + 3.95i)16-s + (−3.49 − 6.05i)17-s + ⋯
L(s)  = 1  + (0.347 − 0.937i)2-s + (−0.499 + 0.288i)3-s + (−0.757 − 0.652i)4-s + (−1.38 − 0.797i)5-s + (0.0966 + 0.569i)6-s + (−0.900 + 0.434i)7-s + (−0.875 + 0.483i)8-s + (0.166 − 0.288i)9-s + (−1.22 + 1.01i)10-s + (1.06 − 0.613i)11-s + (0.567 + 0.107i)12-s − 0.399i·13-s + (0.0936 + 0.995i)14-s + 0.920·15-s + (0.148 + 0.988i)16-s + (−0.847 − 1.46i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.999 - 0.0298i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ -0.999 - 0.0298i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00779276 + 0.521785i\)
\(L(\frac12)\) \(\approx\) \(0.00779276 + 0.521785i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.491 + 1.32i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (2.38 - 1.14i)T \)
good5 \( 1 + (3.08 + 1.78i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.52 + 2.03i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.44iT - 13T^{2} \)
17 \( 1 + (3.49 + 6.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.261 - 0.150i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.21 + 2.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.151iT - 29T^{2} \)
31 \( 1 + (-2.37 - 4.11i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (9.82 + 5.67i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.239T + 41T^{2} \)
43 \( 1 - 1.32iT - 43T^{2} \)
47 \( 1 + (-3.17 + 5.50i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.18 + 2.99i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.73 - 5.62i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.64 - 2.10i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.79 + 2.19i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.46T + 71T^{2} \)
73 \( 1 + (0.284 + 0.493i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.746 + 1.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.0iT - 83T^{2} \)
89 \( 1 + (-1.83 + 3.17i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−12.01301459545031880271766272231, −11.67060102078579675745434928661, −10.56336624894950578322717085374, −9.205533942633148291121417193360, −8.691482394641041476638335830340, −6.83652126375744185946368379461, −5.38615931287336467305464496846, −4.28155912932974227032764706727, −3.24442379993320311174409858480, −0.47127623184728739407868778741, 3.58522956799342535365465834942, 4.36580169533897964709862326947, 6.32231392737152931530511130136, 6.84946122635399429576862095390, 7.72355392275169682138980698634, 8.994874370107079292274524701327, 10.38442371292043131335034877095, 11.60402436892319049936880755500, 12.33180887760478076224452935458, 13.30202130087301765787523497776

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