Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 7 $
Sign $0.764 + 0.644i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.71 − 0.247i)3-s + (−1.28 − 2.23i)5-s + (−0.203 − 2.63i)7-s + (2.87 − 0.849i)9-s + (1.43 + 0.826i)11-s + 5.71i·13-s + (−2.76 − 3.50i)15-s + (−3.79 + 6.56i)17-s + (2.58 − 1.49i)19-s + (−1.00 − 4.47i)21-s + (0.249 − 0.143i)23-s + (−0.825 + 1.43i)25-s + (4.72 − 2.16i)27-s + 2.05i·29-s + (−5.21 − 3.00i)31-s + ⋯
L(s)  = 1  + (0.989 − 0.142i)3-s + (−0.576 − 0.998i)5-s + (−0.0768 − 0.997i)7-s + (0.959 − 0.283i)9-s + (0.431 + 0.249i)11-s + 1.58i·13-s + (−0.713 − 0.906i)15-s + (−0.919 + 1.59i)17-s + (0.594 − 0.343i)19-s + (−0.218 − 0.975i)21-s + (0.0519 − 0.0300i)23-s + (−0.165 + 0.286i)25-s + (0.908 − 0.417i)27-s + 0.382i·29-s + (−0.936 − 0.540i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $0.764 + 0.644i$
motivic weight  =  \(1\)
character  :  $\chi_{168} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 168,\ (\ :1/2),\ 0.764 + 0.644i)\)
\(L(1)\)  \(\approx\)  \(1.33449 - 0.487149i\)
\(L(\frac12)\)  \(\approx\)  \(1.33449 - 0.487149i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.71 + 0.247i)T \)
7 \( 1 + (0.203 + 2.63i)T \)
good5 \( 1 + (1.28 + 2.23i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.43 - 0.826i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.71iT - 13T^{2} \)
17 \( 1 + (3.79 - 6.56i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.58 + 1.49i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.249 + 0.143i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.05iT - 29T^{2} \)
31 \( 1 + (5.21 + 3.00i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.877 + 1.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.28T + 41T^{2} \)
43 \( 1 - 2.46T + 43T^{2} \)
47 \( 1 + (-0.186 - 0.323i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.73 - 3.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.89 - 8.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.889 + 0.513i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.18 - 2.04i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.6iT - 71T^{2} \)
73 \( 1 + (3.30 + 1.90i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.56 - 7.89i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.65T + 83T^{2} \)
89 \( 1 + (7.25 + 12.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.43iT - 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.84896978802794153022324204706, −11.88122323425510496001838306861, −10.63208779275553374627134498588, −9.301760711895233801426609738924, −8.716371625709911432839118946288, −7.57860854879588290216151685740, −6.64503237497898932631942645424, −4.45112170917309975709588899083, −3.88520029547800311132316240542, −1.63347844976968252628310885627, 2.65262409796952525560170474544, 3.47095154039627990587849372921, 5.24603651174442853717034333817, 6.84092029882153306774021482580, 7.78453213287741594126384367849, 8.808740373815350328023566286470, 9.774945092992302837304795147231, 10.90802873316249977848292585622, 11.87788529413406588984770904220, 13.00745690949912017116724281664

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