Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 7 $
Sign $0.212 + 0.977i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.45 + 0.934i)3-s + (−1.90 − 3.29i)5-s + (2.23 − 1.41i)7-s + (1.25 − 2.72i)9-s + (−0.309 − 0.178i)11-s − 4.04i·13-s + (5.84 + 3.02i)15-s + (−0.0519 + 0.0900i)17-s + (−2.12 + 1.22i)19-s + (−1.93 + 4.15i)21-s + (1.15 − 0.665i)23-s + (−4.72 + 8.17i)25-s + (0.723 + 5.14i)27-s − 4.97i·29-s + (−6.83 − 3.94i)31-s + ⋯
L(s)  = 1  + (−0.841 + 0.539i)3-s + (−0.849 − 1.47i)5-s + (0.844 − 0.535i)7-s + (0.417 − 0.908i)9-s + (−0.0933 − 0.0538i)11-s − 1.12i·13-s + (1.50 + 0.780i)15-s + (−0.0126 + 0.0218i)17-s + (−0.487 + 0.281i)19-s + (−0.422 + 0.906i)21-s + (0.240 − 0.138i)23-s + (−0.944 + 1.63i)25-s + (0.139 + 0.990i)27-s − 0.923i·29-s + (−1.22 − 0.708i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $0.212 + 0.977i$
motivic weight  =  \(1\)
character  :  $\chi_{168} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 168,\ (\ :1/2),\ 0.212 + 0.977i)\)
\(L(1)\)  \(\approx\)  \(0.580431 - 0.467637i\)
\(L(\frac12)\)  \(\approx\)  \(0.580431 - 0.467637i\)
\(L(\frac{3}{2})\)   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 \( 1 + (1.45 - 0.934i)T \)
7 \( 1 + (-2.23 + 1.41i)T \)
good5 \( 1 + (1.90 + 3.29i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.309 + 0.178i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.04iT - 13T^{2} \)
17 \( 1 + (0.0519 - 0.0900i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.12 - 1.22i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.15 + 0.665i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.97iT - 29T^{2} \)
31 \( 1 + (6.83 + 3.94i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.45 - 9.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.15T + 41T^{2} \)
43 \( 1 - 0.502T + 43T^{2} \)
47 \( 1 + (-5.72 - 9.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.08 - 2.93i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.77 + 6.53i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.20 + 4.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.34 - 2.32i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.78iT - 71T^{2} \)
73 \( 1 + (0.203 + 0.117i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.61 + 2.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.07T + 83T^{2} \)
89 \( 1 + (-3.41 - 5.90i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.14iT - 97T^{2} \)
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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

−12.49523311984808321447377245759, −11.54446826587137458125053975180, −10.80360064336434234602182699877, −9.607514709529204214783172446154, −8.392399409407288327698312753786, −7.59984695944053419738510150834, −5.78519650436450316121937513743, −4.78221323829187724736595166381, −3.99978110978978369908012586460, −0.817357482609938198999289428208, 2.26273609731735888471291721841, 4.14305091840511615067589918876, 5.60823087822596602587681234578, 6.92144614382974736518841851602, 7.42269395832198513685340709698, 8.781600578030682514082250459444, 10.51036574552914769686450444955, 11.21065545985809241858736414191, 11.71823853056333805883833604012, 12.75919379931219567667696243102

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