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  :  $\chi_{168} (89, \cdot )$
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

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

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