Properties

Degree 2
Conductor $ 2 \cdot 3^{4} \cdot 7 $
Sign $0.766 - 0.642i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (2 − 3.46i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s + 6·17-s + 2·19-s + (2.5 + 4.33i)25-s + 3.99·26-s + 0.999·28-s + (3 + 5.19i)29-s + (2 − 3.46i)31-s + (0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.188 − 0.327i)7-s − 0.353·8-s + (0.554 − 0.960i)13-s + (0.133 − 0.231i)14-s + (−0.125 − 0.216i)16-s + 1.45·17-s + 0.458·19-s + (0.5 + 0.866i)25-s + 0.784·26-s + 0.188·28-s + (0.557 + 0.964i)29-s + (0.359 − 0.622i)31-s + (0.0883 − 0.153i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
\( \varepsilon \)  =  $0.766 - 0.642i$
motivic weight  =  \(1\)
character  :  $\chi_{1134} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1134,\ (\ :1/2),\ 0.766 - 0.642i)$
$L(1)$  $\approx$  $1.981341956$
$L(\frac12)$  $\approx$  $1.981341956$
$L(\frac{3}{2})$   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 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.896832065968700209378951010560, −8.981262373361538721293537345588, −8.031293774802245118875663112020, −7.48655076274356680955745994755, −6.51934795054915739147487627171, −5.63891753560402551551713279035, −4.95184129186987004969716310671, −3.67027232218535153757676445394, −3.01118174722901517185218105187, −1.06526607354851973657984272848, 1.10167326553198431223830906085, 2.43237795903141877385950354742, 3.45173581884622919931806138410, 4.38023679322460222737079841298, 5.40485379215751779468256095872, 6.20692714034726877009450547907, 7.13726025543389462193913449045, 8.294617437855192102100987822451, 8.974194983815212186385065008631, 9.965363900033757056542797857688

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