Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} $
Sign $0.426 - 0.904i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.27 + 1.89i)11-s + 8.02i·17-s + 8.34·19-s + (2.5 + 4.33i)25-s + (0.398 + 0.230i)41-s + (−1.17 − 2.03i)43-s + (−3.5 + 6.06i)49-s + (10.6 + 6.13i)59-s + (−7.17 + 12.4i)67-s + 13.6·73-s + (−2.44 + 1.41i)83-s − 5.65i·89-s + (−9.84 − 17.0i)97-s + 4.70i·107-s + (−9.79 − 5.65i)113-s + ⋯
L(s)  = 1  + (−0.987 + 0.570i)11-s + 1.94i·17-s + 1.91·19-s + (0.5 + 0.866i)25-s + (0.0623 + 0.0359i)41-s + (−0.179 − 0.310i)43-s + (−0.5 + 0.866i)49-s + (1.38 + 0.798i)59-s + (−0.876 + 1.51i)67-s + 1.60·73-s + (−0.268 + 0.155i)83-s − 0.599i·89-s + (−0.999 − 1.73i)97-s + 0.454i·107-s + (−0.921 − 0.532i)113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(864\)    =    \(2^{5} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.426 - 0.904i$
motivic weight  =  \(1\)
character  :  $\chi_{864} (719, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 864,\ (\ :1/2),\ 0.426 - 0.904i)$
$L(1)$  $\approx$  $1.15004 + 0.728934i$
$L(\frac12)$  $\approx$  $1.15004 + 0.728934i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.27 - 1.89i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 8.02iT - 17T^{2} \)
19 \( 1 - 8.34T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-0.398 - 0.230i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.17 + 2.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-10.6 - 6.13i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.17 - 12.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 13.6T + 73T^{2} \)
79 \( 1 + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.44 - 1.41i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.65iT - 89T^{2} \)
97 \( 1 + (9.84 + 17.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.28466486575814716285276580168, −9.596197586606250809916012474307, −8.564993072904004637242555315115, −7.75356880050843623520001103169, −7.02633632569804065955786419058, −5.82253065247193381760125788530, −5.12405992319947951324973678043, −3.93594543018270334249324761742, −2.85814276036973291429470768753, −1.47204781867105401788957432602, 0.70359380349248619104052092130, 2.56558931867971276934818502418, 3.37572739226034554294294729497, 4.92248920510455505575774375336, 5.38808268541079383633326944895, 6.65578894929035871887264689745, 7.51289408425471608124750486816, 8.226139525429808086681492738152, 9.322061658197877863940058419151, 9.894698935907858296858069353309

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