Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 7 $
Sign $-0.866 - 0.499i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.21 + 1.22i)3-s + (−1.40 + 2.43i)5-s + (−2.08 − 1.62i)7-s + (−0.0238 − 2.99i)9-s + (−4.74 + 2.74i)11-s + 1.35i·13-s + (−1.27 − 4.69i)15-s + (2.88 + 5.00i)17-s + (1.71 + 0.992i)19-s + (4.54 − 0.579i)21-s + (2.09 + 1.21i)23-s + (−1.44 − 2.49i)25-s + (3.71 + 3.63i)27-s − 7.05i·29-s + (−3.07 + 1.77i)31-s + ⋯
L(s)  = 1  + (−0.704 + 0.709i)3-s + (−0.627 + 1.08i)5-s + (−0.788 − 0.615i)7-s + (−0.00795 − 0.999i)9-s + (−1.43 + 0.826i)11-s + 0.376i·13-s + (−0.329 − 1.21i)15-s + (0.700 + 1.21i)17-s + (0.394 + 0.227i)19-s + (0.991 − 0.126i)21-s + (0.437 + 0.252i)23-s + (−0.288 − 0.499i)25-s + (0.715 + 0.698i)27-s − 1.31i·29-s + (−0.552 + 0.318i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $-0.866 - 0.499i$
motivic weight  =  \(1\)
character  :  $\chi_{168} (89, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 168,\ (\ :1/2),\ -0.866 - 0.499i)\)
\(L(1)\)  \(\approx\)  \(0.128791 + 0.480776i\)
\(L(\frac12)\)  \(\approx\)  \(0.128791 + 0.480776i\)
\(L(\frac{3}{2})\)   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 \( 1 + (1.21 - 1.22i)T \)
7 \( 1 + (2.08 + 1.62i)T \)
good5 \( 1 + (1.40 - 2.43i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.74 - 2.74i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.35iT - 13T^{2} \)
17 \( 1 + (-2.88 - 5.00i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.71 - 0.992i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.09 - 1.21i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.05iT - 29T^{2} \)
31 \( 1 + (3.07 - 1.77i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.14 - 3.71i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.81T + 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + (0.201 - 0.348i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.28 - 3.04i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.28 - 2.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.75 + 2.74i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.45 - 5.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.08iT - 71T^{2} \)
73 \( 1 + (0.295 - 0.170i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.19 + 2.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + (0.576 - 0.998i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.0iT - 97T^{2} \)
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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

−13.02189075105338747768138649530, −12.12356357336383194579016336356, −10.93138462953627401086163953151, −10.39190421281249939339701453135, −9.615870259833836478215065935794, −7.79654837456688486563625304802, −6.91249305199321856535848625322, −5.74174983219573723457030630915, −4.23907614243485503300542091550, −3.15169494529441115946275292868, 0.49503094609902077882147536377, 2.94124069348704167821615651012, 5.06697063244296710843847252222, 5.65913838288105539841790727484, 7.22133174025367883757898676071, 8.147231868546498121494030643569, 9.182414199763645268414004668611, 10.59712744731384832730724131405, 11.59532750932295647688508743935, 12.62289298758542205272319297801

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