Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 7 $
Sign $0.895 + 0.444i$
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)3-s + (−0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s − 13-s + (0.5 − 0.866i)19-s + 0.999·21-s + (−0.5 − 0.866i)25-s + 0.999·27-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)39-s − 43-s + (−0.499 − 0.866i)49-s − 0.999·57-s + (−1 + 1.73i)61-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s − 13-s + (0.5 − 0.866i)19-s + 0.999·21-s + (−0.5 − 0.866i)25-s + 0.999·27-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)39-s − 43-s + (−0.499 − 0.866i)49-s − 0.999·57-s + (−1 + 1.73i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $0.895 + 0.444i$
motivic weight  =  \(0\)
character  :  $\chi_{84} (65, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 84,\ (\ :0),\ 0.895 + 0.444i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.4550711842\)
\(L(\frac12)\)  \(\approx\)  \(0.4550711842\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) 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 \( 1 \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - 2T + T^{2} \)
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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

−14.33197631703146060715751911730, −13.20498120399190008275121892141, −12.27926998190461195602564601842, −11.57081641310592920438165715557, −10.11348932277312344731628512468, −8.796010299104171695587299127572, −7.44725460239841168931737259170, −6.30895696473354994681334896900, −5.07920460323713432335456360487, −2.56497434085969418732679777437, 3.52084052442906496218604031148, 4.89098280278214127535463544377, 6.33268371238830377081639818438, 7.74268924650426005669373898019, 9.514310179830675798001626081592, 10.10977840826758241325303230439, 11.28616606617349332842095808165, 12.33651690769049716182029767608, 13.63570593099300485737171669080, 14.74600246740631530719188745074

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