Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.249 - 0.968i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 1.11i)2-s + (−0.500 − 1.93i)4-s + 2.23i·5-s + 3.87i·7-s + (2.59 + 1.11i)8-s + (−2.50 − 1.93i)10-s − 1.73·11-s + 2·13-s + (−4.33 − 3.35i)14-s + (−3.5 + 1.93i)16-s − 4.47i·17-s + (4.33 − 1.11i)20-s + (1.49 − 1.93i)22-s + 6.92·23-s + (−1.73 + 2.23i)26-s + ⋯
L(s)  = 1  + (−0.612 + 0.790i)2-s + (−0.250 − 0.968i)4-s + 0.999i·5-s + 1.46i·7-s + (0.918 + 0.395i)8-s + (−0.790 − 0.612i)10-s − 0.522·11-s + 0.554·13-s + (−1.15 − 0.896i)14-s + (−0.875 + 0.484i)16-s − 1.08i·17-s + (0.968 − 0.250i)20-s + (0.319 − 0.412i)22-s + 1.44·23-s + (−0.339 + 0.438i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.249 - 0.968i$
motivic weight  =  \(1\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 108,\ (\ :1/2),\ -0.249 - 0.968i)$
$L(1)$  $\approx$  $0.454982 + 0.587379i$
$L(\frac12)$  $\approx$  $0.454982 + 0.587379i$
$L(\frac{3}{2})$   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 \( 1 + (0.866 - 1.11i)T \)
3 \( 1 \)
good5 \( 1 - 2.23iT - 5T^{2} \)
7 \( 1 - 3.87iT - 7T^{2} \)
11 \( 1 + 1.73T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 4.47iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 6.92T + 23T^{2} \)
29 \( 1 + 4.47iT - 29T^{2} \)
31 \( 1 + 3.87iT - 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 - 8.94iT - 41T^{2} \)
43 \( 1 + 7.74iT - 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 - 2.23iT - 53T^{2} \)
59 \( 1 + 3.46T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 - 7.74iT - 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - 5T + 73T^{2} \)
79 \( 1 + 7.74iT - 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + 4.47iT - 89T^{2} \)
97 \( 1 - 11T + 97T^{2} \)
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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

−14.33053538956708208078913437617, −13.22044323129279074163582784380, −11.65860914094312705145890406307, −10.74305424989269636079212311014, −9.519585553092712041510098831019, −8.605039139736456510941004027840, −7.35020195338513262000920046682, −6.24844149227236302898293430355, −5.16674222557969672828772180192, −2.66172949639314995069071382847, 1.18580599730778824673139974474, 3.59964926771827505759024991550, 4.85330436059790058789154442370, 7.03079453687674482395396872795, 8.201124739732859630903453769177, 9.102591024841214523748642571346, 10.45275716925881041819960738904, 10.96576914492129241011467934779, 12.53064887967947458407139524201, 13.06454789991610280134393753490

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