Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} $
Sign $-0.710 + 0.703i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.72 − 3.30i)11-s − 2.36i·17-s − 6.34·19-s + (2.5 − 4.33i)25-s + (−9.39 + 5.42i)41-s + (6.17 − 10.6i)43-s + (−3.5 − 6.06i)49-s + (−1.62 + 0.937i)59-s + (0.174 + 0.301i)67-s − 15.6·73-s + (2.44 + 1.41i)83-s − 5.65i·89-s + (4.84 − 8.39i)97-s + 15.0i·107-s + (9.79 − 5.65i)113-s + ⋯
L(s)  = 1  + (−1.72 − 0.996i)11-s − 0.574i·17-s − 1.45·19-s + (0.5 − 0.866i)25-s + (−1.46 + 0.847i)41-s + (0.941 − 1.63i)43-s + (−0.5 − 0.866i)49-s + (−0.211 + 0.122i)59-s + (0.0212 + 0.0368i)67-s − 1.83·73-s + (0.268 + 0.155i)83-s − 0.599i·89-s + (0.492 − 0.852i)97-s + 1.45i·107-s + (0.921 − 0.532i)113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(864\)    =    \(2^{5} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.710 + 0.703i$
motivic weight  =  \(1\)
character  :  $\chi_{864} (143, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 864,\ (\ :1/2),\ -0.710 + 0.703i)$
$L(1)$  $\approx$  $0.241322 - 0.586427i$
$L(\frac12)$  $\approx$  $0.241322 - 0.586427i$
$L(\frac{3}{2})$   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 \( 1 + (5.72 + 3.30i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.36iT - 17T^{2} \)
19 \( 1 + 6.34T + 19T^{2} \)
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 \( 1 + (9.39 - 5.42i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.17 + 10.6i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (1.62 - 0.937i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.174 - 0.301i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
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 \( 1 + (-4.84 + 8.39i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.10841869937682695523366732733, −8.751378000354444505871073096281, −8.316785505147263982161513656938, −7.34369921565016040548859540005, −6.32366234647603984538956438741, −5.42758390908572117678234929913, −4.55030640314409920301380996519, −3.22058577297882664406571934870, −2.27140139482934159744990179189, −0.28322000593794535220379061420, 1.88199120633321272955600922752, 2.94822875901948585535418123220, 4.32755069570636227057242602629, 5.11748741138339969180679279658, 6.13232771009769856770193737958, 7.16963759193836912097395500915, 7.922729258291229715183688680598, 8.723320361851511402446308445901, 9.767288433596111259394088210102, 10.52476942392600182470673747046

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