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  :  $\chi_{378} (211, \cdot )$
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

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

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