Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 7 $
Sign $-0.364 - 0.931i$
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 + (−1.87 + 3.23i)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 + (−0.500 + 0.865i)14-s + (0.258 − 0.577i)15-s + (−0.875 − 0.484i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $-0.364 - 0.931i$
motivic weight  =  \(1\)
character  :  $\chi_{168} (125, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 168,\ (\ :1/2),\ -0.364 - 0.931i)\)
\(L(1)\)  \(\approx\)  \(0.632989 + 0.927867i\)
\(L(\frac12)\)  \(\approx\)  \(0.632989 + 0.927867i\)
\(L(\frac{3}{2})\)   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

−13.11660383068736666303688746960, −12.20885634828145972281263271368, −11.39452285114443737005414732863, −10.37012956915076619494846843637, −8.634933315305632660782581472103, −7.67758562763704143515888742733, −6.54615182851605799281723944998, −5.70900835221872772173934723285, −4.71718816802015742626785186484, −2.76039720214996990031075185995, 1.08963056607366955327383653957, 3.57090048957578998834165564553, 4.79457472372657306113581235709, 5.49725246135198787427274720579, 6.94893391405148975373116210242, 8.622653182154453374979134000252, 10.00297039065601042563891456515, 10.61703020774447213663370149404, 11.47486149676968314924475347063, 12.39712138888186013090680942096

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