Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} $
Sign $-0.737 - 0.675i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 − 1.36i)2-s + (−1.5 + 0.866i)3-s + (−1.73 − i)4-s + (−3.73 + i)5-s + (0.633 + 2.36i)6-s + (−0.633 + 0.366i)7-s + (−2 + 1.99i)8-s + (1.5 − 2.59i)9-s + 5.46i·10-s + (−0.767 + 2.86i)11-s + 3.46·12-s + (−1.63 − 6.09i)13-s + (0.267 + i)14-s + (4.73 − 4.73i)15-s + (1.99 + 3.46i)16-s − 2.26·17-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 + 0.499i)3-s + (−0.866 − 0.5i)4-s + (−1.66 + 0.447i)5-s + (0.258 + 0.965i)6-s + (−0.239 + 0.138i)7-s + (−0.707 + 0.707i)8-s + (0.5 − 0.866i)9-s + 1.72i·10-s + (−0.231 + 0.864i)11-s + 0.999·12-s + (−0.453 − 1.69i)13-s + (0.0716 + 0.267i)14-s + (1.22 − 1.22i)15-s + (0.499 + 0.866i)16-s − 0.550·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(144\)    =    \(2^{4} \cdot 3^{2}\)
\( \varepsilon \)  =  $-0.737 - 0.675i$
motivic weight  =  \(1\)
character  :  $\chi_{144} (85, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 144,\ (\ :1/2),\ -0.737 - 0.675i)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.366 + 1.36i)T \)
3 \( 1 + (1.5 - 0.866i)T \)
good5 \( 1 + (3.73 - i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.633 - 0.366i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.767 - 2.86i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.63 + 6.09i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 2.26T + 17T^{2} \)
19 \( 1 + (0.633 - 0.633i)T - 19iT^{2} \)
23 \( 1 + (1.09 + 0.633i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.36 + 0.633i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (3.73 - 6.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.26 - 1.26i)T + 37iT^{2} \)
41 \( 1 + (2.59 + 1.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.330 + 1.23i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (4.83 + 8.36i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.535 + 0.535i)T + 53iT^{2} \)
59 \( 1 + (-4.96 + 1.33i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (3 + 0.803i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.40 - 5.23i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 - 9.73iT - 73T^{2} \)
79 \( 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.36 + 0.366i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 2iT - 89T^{2} \)
97 \( 1 + (4.13 + 7.16i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.35406393199146470249704748200, −11.45372218435714085317694819586, −10.65168405786904703904659716101, −9.914292322835849841623360863220, −8.374146813639714627419601638633, −7.06835395182399365715191704594, −5.37154499169052242612325972392, −4.28460686629518984921253603521, −3.18789624094797918063610567022, 0, 3.94487810149689928097722800912, 4.85893214325361573220573028109, 6.33451094018827002275238546269, 7.28584992209736387105254906852, 8.106812078482140624736823041580, 9.259790703264264492374033140900, 11.17571947774232940917086952977, 11.79235197799433396881266469204, 12.73723088764880782990822676101

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