Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 1.11i)2-s + (1.58 − 0.707i)3-s + (−0.500 − 1.93i)4-s + 1.41i·5-s + (0.578 − 2.38i)6-s + (1 + 2.44i)7-s + (−2.59 − 1.11i)8-s + (2.00 − 2.23i)9-s + (1.58 + 1.22i)10-s − 3.46·11-s + (−2.15 − 2.70i)12-s − 3.16·13-s + (3.60 + 1.00i)14-s + (1.00 + 2.23i)15-s + (−3.5 + 1.93i)16-s + ⋯
L(s)  = 1  + (0.612 − 0.790i)2-s + (0.912 − 0.408i)3-s + (−0.250 − 0.968i)4-s + 0.632i·5-s + (0.236 − 0.971i)6-s + (0.377 + 0.925i)7-s + (−0.918 − 0.395i)8-s + (0.666 − 0.745i)9-s + (0.500 + 0.387i)10-s − 1.04·11-s + (−0.623 − 0.781i)12-s − 0.877·13-s + (0.963 + 0.268i)14-s + (0.258 + 0.577i)15-s + (−0.875 + 0.484i)16-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $0.391 + 0.920i$
motivic weight  =  \(1\)
character  :  $\chi_{168} (125, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 168,\ (\ :1/2),\ 0.391 + 0.920i)\)
\(L(1)\)  \(\approx\)  \(1.53953 - 1.01863i\)
\(L(\frac12)\)  \(\approx\)  \(1.53953 - 1.01863i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.866 + 1.11i)T \)
3 \( 1 + (-1.58 + 0.707i)T \)
7 \( 1 + (-1 - 2.44i)T \)
good5 \( 1 - 1.41iT - 5T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 + 3.16T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 3.16T + 19T^{2} \)
23 \( 1 - 4.47iT - 23T^{2} \)
29 \( 1 - 6.92T + 29T^{2} \)
31 \( 1 + 4.89iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 + 7.74iT - 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 9.89iT - 59T^{2} \)
61 \( 1 + 3.16T + 61T^{2} \)
67 \( 1 - 7.74iT - 67T^{2} \)
71 \( 1 + 8.94iT - 71T^{2} \)
73 \( 1 + 14.6iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + 7.07iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 4.89iT - 97T^{2} \)
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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

−12.65532777958713057365053791591, −11.84732355866066343089509768712, −10.67234043449732680498904501162, −9.715948519676351064297965486558, −8.648562051946870806168979238225, −7.43612670772046266191529442110, −6.05217269473007950929433294918, −4.68690920803570552858768595132, −3.00992148967276773330653814579, −2.21259845014422628275842591264, 2.80169869128384198928605592664, 4.41750777975697444234847449990, 4.95793576867031106065680223806, 6.82858601732665523858397875195, 7.928272798620253663184366360882, 8.497790944170706593788270040492, 9.811813870174425823366423492081, 10.88905315844534285904416147144, 12.60790505601967950142467460542, 13.03620235591779926882153435974

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