Properties

Degree 2
Conductor $ 2 \cdot 3^{3} \cdot 7 $
Sign $0.129 - 0.991i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.766 + 0.642i)2-s + (1.23 + 1.21i)3-s + (0.173 + 0.984i)4-s + (0.667 + 0.243i)5-s + (0.159 + 1.72i)6-s + (0.173 − 0.984i)7-s + (−0.500 + 0.866i)8-s + (0.0308 + 2.99i)9-s + (0.355 + 0.615i)10-s + (0.297 − 0.108i)11-s + (−0.986 + 1.42i)12-s + (−0.797 + 0.669i)13-s + (0.766 − 0.642i)14-s + (0.525 + 1.11i)15-s + (−0.939 + 0.342i)16-s + (0.219 + 0.380i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (0.710 + 0.703i)3-s + (0.0868 + 0.492i)4-s + (0.298 + 0.108i)5-s + (0.0652 + 0.704i)6-s + (0.0656 − 0.372i)7-s + (−0.176 + 0.306i)8-s + (0.0102 + 0.999i)9-s + (0.112 + 0.194i)10-s + (0.0897 − 0.0326i)11-s + (−0.284 + 0.411i)12-s + (−0.221 + 0.185i)13-s + (0.204 − 0.171i)14-s + (0.135 + 0.287i)15-s + (−0.234 + 0.0855i)16-s + (0.0533 + 0.0923i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.129 - 0.991i$
motivic weight  =  \(1\)
character  :  $\chi_{378} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 378,\ (\ :1/2),\ 0.129 - 0.991i)$
$L(1)$  $\approx$  $1.72036 + 1.51002i$
$L(\frac12)$  $\approx$  $1.72036 + 1.51002i$
$L(\frac{3}{2})$   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 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (-1.23 - 1.21i)T \)
7 \( 1 + (-0.173 + 0.984i)T \)
good5 \( 1 + (-0.667 - 0.243i)T + (3.83 + 3.21i)T^{2} \)
11 \( 1 + (-0.297 + 0.108i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (0.797 - 0.669i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.219 - 0.380i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.10 + 3.64i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0614 + 0.348i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-0.815 - 0.684i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.00644 - 0.0365i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (3.45 + 5.99i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.76 + 1.47i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-2.77 + 1.00i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.124 - 0.705i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + 2.66T + 53T^{2} \)
59 \( 1 + (-11.9 - 4.34i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-2.13 + 12.0i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (7.32 - 6.14i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-3.46 - 6.00i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.26 + 9.12i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.98 + 7.54i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-3.97 - 3.33i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (5.34 - 9.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.98 + 1.45i)T + (74.3 - 62.3i)T^{2} \)
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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

−11.52305685028421830012005521553, −10.58935060169884909980362640206, −9.664462477989729912257162504069, −8.805828268444889601534523122895, −7.79576717203350950510803429173, −6.90329592610163468924544332137, −5.60287343377745765685259605805, −4.59176112734162744810896541374, −3.62192364191525133828972228113, −2.36365321909676315775877685512, 1.50713166035009154506417130709, 2.71664299695834535069410245037, 3.83777188474594172286123159665, 5.31992600860724170088032223681, 6.27223176449016255095409668261, 7.40093049890554666365044047575, 8.379639705465587326212122528728, 9.409744706454045141422913204957, 10.12994279884193078521898153433, 11.49834875268161616506768173385

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