Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 7 $
Sign $0.987 + 0.155i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.62 − 0.601i)3-s + (0.0726 + 0.125i)5-s + (1.05 + 2.42i)7-s + (2.27 − 1.95i)9-s + (−2.13 − 1.23i)11-s − 2.04i·13-s + (0.193 + 0.160i)15-s + (0.878 − 1.52i)17-s + (−3.68 + 2.12i)19-s + (3.17 + 3.30i)21-s + (−7.46 + 4.30i)23-s + (2.48 − 4.31i)25-s + (2.52 − 4.54i)27-s + 7.08i·29-s + (3.11 + 1.80i)31-s + ⋯
L(s)  = 1  + (0.937 − 0.347i)3-s + (0.0324 + 0.0562i)5-s + (0.398 + 0.917i)7-s + (0.758 − 0.651i)9-s + (−0.644 − 0.372i)11-s − 0.566i·13-s + (0.0500 + 0.0414i)15-s + (0.213 − 0.369i)17-s + (−0.844 + 0.487i)19-s + (0.692 + 0.721i)21-s + (−1.55 + 0.898i)23-s + (0.497 − 0.862i)25-s + (0.485 − 0.874i)27-s + 1.31i·29-s + (0.560 + 0.323i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $0.987 + 0.155i$
motivic weight  =  \(1\)
character  :  $\chi_{168} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 168,\ (\ :1/2),\ 0.987 + 0.155i)\)
\(L(1)\)  \(\approx\)  \(1.49845 - 0.116921i\)
\(L(\frac12)\)  \(\approx\)  \(1.49845 - 0.116921i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.62 + 0.601i)T \)
7 \( 1 + (-1.05 - 2.42i)T \)
good5 \( 1 + (-0.0726 - 0.125i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.13 + 1.23i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.04iT - 13T^{2} \)
17 \( 1 + (-0.878 + 1.52i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.68 - 2.12i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.46 - 4.30i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.08iT - 29T^{2} \)
31 \( 1 + (-3.11 - 1.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.93 + 5.08i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.33T + 41T^{2} \)
43 \( 1 + 9.19T + 43T^{2} \)
47 \( 1 + (-4.65 - 8.05i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.49 + 2.59i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.60 + 9.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.66 + 2.69i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.57 + 4.45i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.79iT - 71T^{2} \)
73 \( 1 + (-11.3 - 6.52i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.86 - 4.95i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.9T + 83T^{2} \)
89 \( 1 + (-4.34 - 7.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.65iT - 97T^{2} \)
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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

−12.71578425065721881811437572988, −12.08291851424609480725974157228, −10.67852423937041252165465614445, −9.623525153371136573190120735837, −8.430942408257370302185555027421, −7.954167103250830845687834783583, −6.47141940835705618985266775715, −5.16340541342841355067030357509, −3.41259078870631410097943477117, −2.09757629818352275877504583079, 2.13815474993861405299726239242, 3.87000755462529653979581323571, 4.81422590930139713854073495025, 6.69806958057561133189156614833, 7.84604440991890739053254594377, 8.599195323880127248548280507797, 9.975738278232458799330084027021, 10.48193579654684014039740179792, 11.82049188130356442524674077389, 13.18027945168394912625400960967

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