Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 7 $
Sign $0.335 + 0.942i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.291 − 1.70i)3-s + (−0.0726 + 0.125i)5-s + (1.05 − 2.42i)7-s + (−2.83 − 0.993i)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 + (−3.83 − 2.50i)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  + (0.168 − 0.985i)3-s + (−0.0324 + 0.0562i)5-s + (0.398 − 0.917i)7-s + (−0.943 − 0.331i)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 + (−0.837 − 0.547i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.335 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $0.335 + 0.942i$
motivic weight  =  \(1\)
character  :  $\chi_{168} (89, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 168,\ (\ :1/2),\ 0.335 + 0.942i)$
$L(1)$  $\approx$  $0.989054 - 0.698025i$
$L(\frac12)$  $\approx$  $0.989054 - 0.698025i$
$L(\frac{3}{2})$   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 \( 1 + (-0.291 + 1.70i)T \)
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

−12.79232073125611430486263013203, −11.47385501381790471140589722416, −10.99971306696821398928915074549, −9.347958973557939235782127299776, −8.429246034939105088410315367716, −7.18019396892624074552642705773, −6.61193947361901245248379337844, −4.92582000289964282460318473036, −3.29050507828690273397047372033, −1.36971516016772276785733922471, 2.55279051297150666047897605751, 4.15666143810256498966314157011, 5.23419401872604950365709302053, 6.46794092876122507741277649571, 8.270912812301320782156557229671, 8.850082463767795008009894670912, 10.00137823169201837085327789109, 10.90432037282213600707525089728, 11.92047194535590734916435573968, 12.86479317589200630764130890197

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