Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 7 $
Sign $0.493 - 0.869i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.67 − 0.441i)3-s + (1.40 + 2.43i)5-s + (−2.08 + 1.62i)7-s + (2.60 + 1.47i)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 + (4.21 − 1.80i)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  + (−0.966 − 0.254i)3-s + (0.627 + 1.08i)5-s + (−0.788 + 0.615i)7-s + (0.869 + 0.493i)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 + (0.919 − 0.393i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.493 - 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.493 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $0.493 - 0.869i$
motivic weight  =  \(1\)
character  :  $\chi_{168} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 168,\ (\ :1/2),\ 0.493 - 0.869i)$
$L(1)$  $\approx$  $0.772504 + 0.450048i$
$L(\frac12)$  $\approx$  $0.772504 + 0.450048i$
$L(\frac{3}{2})$   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 \( 1 + (1.67 + 0.441i)T \)
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

−12.76783219782184259959224189862, −12.00654234446597682394779334486, −10.97266565077193816614853574349, −10.06713589418775907777735006691, −9.221856710576984989159257336114, −7.36413984098407004932304082114, −6.38117604804195252677544804280, −5.89363648135415347138813107555, −4.03540025147548806552119052147, −2.15084230867860230027899069378, 1.04060118499167186243482119534, 3.82682366036790885042336687141, 5.02271833642966795620650288408, 6.16805290630852140360322635944, 7.03373781709739093808566228097, 9.040442449523770344223601996133, 9.432050796436887581358766099465, 10.67883892884511582079087487378, 11.71147936156489447828661915572, 12.53574466233292106455996805229

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