Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{2} $
Sign $0.173 + 0.984i$
Motivic weight 0
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 + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.499 + 0.866i)6-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + 0.999·12-s + (−0.5 − 0.866i)16-s − 17-s + 0.999·18-s − 19-s + (0.499 − 0.866i)22-s + (−0.5 − 0.866i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.499 + 0.866i)6-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + 0.999·12-s + (−0.5 − 0.866i)16-s − 17-s + 0.999·18-s − 19-s + (0.499 − 0.866i)22-s + (−0.5 − 0.866i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(72\)    =    \(2^{3} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.173 + 0.984i$
motivic weight  =  \(0\)
character  :  $\chi_{72} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 72,\ (\ :0),\ 0.173 + 0.984i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.3544866523\)
\(L(\frac12)\)  \(\approx\)  \(0.3544866523\)
\(L(1)\)   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 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−14.34571041592205361528943903242, −13.14488498825266391224630030241, −12.39764694798549826442099294177, −11.43543784024726873689769103533, −10.42530837244236248432447086564, −9.026004063438331298527805431678, −7.78292414143922961147563368342, −6.52229259023448238669588328170, −4.47388984368426759808121563855, −2.13460287732343082794920450963, 4.17397350823654265270160808926, 5.63538510576822836356840705680, 6.70640051462313961974619854325, 8.479371373944274566855562461374, 9.352510038710622356505889286642, 10.58013683892231668704707658551, 11.46457120346775008809481426109, 13.25301324941013762299050072315, 14.49356786122127795921995434996, 15.34043328614189626901675784065

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