Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 7 $
Sign $0.920 + 0.391i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0805 − 1.73i)3-s + (1.90 + 3.29i)5-s + (2.23 − 1.41i)7-s + (−2.98 − 0.278i)9-s + (0.309 + 0.178i)11-s − 4.04i·13-s + (5.84 − 3.02i)15-s + (0.0519 − 0.0900i)17-s + (−2.12 + 1.22i)19-s + (−2.26 − 3.98i)21-s + (−1.15 + 0.665i)23-s + (−4.72 + 8.17i)25-s + (−0.723 + 5.14i)27-s + 4.97i·29-s + (−6.83 − 3.94i)31-s + ⋯
L(s)  = 1  + (0.0465 − 0.998i)3-s + (0.849 + 1.47i)5-s + (0.844 − 0.535i)7-s + (−0.995 − 0.0929i)9-s + (0.0933 + 0.0538i)11-s − 1.12i·13-s + (1.50 − 0.780i)15-s + (0.0126 − 0.0218i)17-s + (−0.487 + 0.281i)19-s + (−0.495 − 0.868i)21-s + (−0.240 + 0.138i)23-s + (−0.944 + 1.63i)25-s + (−0.139 + 0.990i)27-s + 0.923i·29-s + (−1.22 − 0.708i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $0.920 + 0.391i$
motivic weight  =  \(1\)
character  :  $\chi_{168} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 168,\ (\ :1/2),\ 0.920 + 0.391i)$
$L(1)$  $\approx$  $1.30542 - 0.266149i$
$L(\frac12)$  $\approx$  $1.30542 - 0.266149i$
$L(\frac{3}{2})$   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 \( 1 + (-0.0805 + 1.73i)T \)
7 \( 1 + (-2.23 + 1.41i)T \)
good5 \( 1 + (-1.90 - 3.29i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.309 - 0.178i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.04iT - 13T^{2} \)
17 \( 1 + (-0.0519 + 0.0900i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.12 - 1.22i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.15 - 0.665i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.97iT - 29T^{2} \)
31 \( 1 + (6.83 + 3.94i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.45 - 9.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.15T + 41T^{2} \)
43 \( 1 - 0.502T + 43T^{2} \)
47 \( 1 + (5.72 + 9.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.08 + 2.93i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.77 - 6.53i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.20 + 4.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.34 - 2.32i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.78iT - 71T^{2} \)
73 \( 1 + (0.203 + 0.117i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.61 + 2.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.07T + 83T^{2} \)
89 \( 1 + (3.41 + 5.90i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.14iT - 97T^{2} \)
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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

−12.98676476614376235510784831787, −11.60008827499422557716955364166, −10.80713039769646776126177312380, −9.972224887704967154345561660082, −8.335216464299551128745421496166, −7.39044511587630937233808389011, −6.52497170698822048212672937751, −5.45109728803194678722569803022, −3.24001845956244916305295914090, −1.87291598327957002758365077659, 1.99561098480782906180991168269, 4.27865416442050184480207527054, 5.06986487046246025043120490336, 6.05220565933164394711759628988, 8.144973828748834967057947891953, 9.069013713756212065036538708000, 9.474144494569531890198705294403, 10.86945006210972323392032594350, 11.81921500272016051120446533419, 12.82288738152559603027188633637

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