Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.816 + 0.577i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.31i·5-s + i·7-s + 4.72·11-s + 4.97·13-s + 0.484i·17-s − 2.29i·19-s + 7.97·23-s − 5.97·25-s − 1.41i·29-s + 7.66i·31-s + 3.31·35-s + 2.39·37-s + 6.55i·41-s + 5.37i·43-s + 6.21·47-s + ⋯
L(s)  = 1  − 1.48i·5-s + 0.377i·7-s + 1.42·11-s + 1.38·13-s + 0.117i·17-s − 0.525i·19-s + 1.66·23-s − 1.19·25-s − 0.262i·29-s + 1.37i·31-s + 0.560·35-s + 0.393·37-s + 1.02i·41-s + 0.819i·43-s + 0.906·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.816 + 0.577i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (575, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.816 + 0.577i)$
$L(1)$  $\approx$  $2.445467761$
$L(\frac12)$  $\approx$  $2.445467761$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) 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 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 + 3.31iT - 5T^{2} \)
11 \( 1 - 4.72T + 11T^{2} \)
13 \( 1 - 4.97T + 13T^{2} \)
17 \( 1 - 0.484iT - 17T^{2} \)
19 \( 1 + 2.29iT - 19T^{2} \)
23 \( 1 - 7.97T + 23T^{2} \)
29 \( 1 + 1.41iT - 29T^{2} \)
31 \( 1 - 7.66iT - 31T^{2} \)
37 \( 1 - 2.39T + 37T^{2} \)
41 \( 1 - 6.55iT - 41T^{2} \)
43 \( 1 - 5.37iT - 43T^{2} \)
47 \( 1 - 6.21T + 47T^{2} \)
53 \( 1 - 1.00iT - 53T^{2} \)
59 \( 1 + 1.38T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 - 3.27iT - 67T^{2} \)
71 \( 1 - 3.34T + 71T^{2} \)
73 \( 1 + 2.10T + 73T^{2} \)
79 \( 1 - 12.0iT - 79T^{2} \)
83 \( 1 - 3.24T + 83T^{2} \)
89 \( 1 - 5.72iT - 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−8.616133793787232728291373940565, −7.83564390756393400708377530215, −6.72377289320838529218248628900, −6.20049234615255610906225957070, −5.28766893528628699347634635323, −4.63159417791880573544708459129, −3.90024403737894514009159396125, −2.94859200500914392048786501511, −1.41086467593643744262265600826, −1.04866839736139433784435951726, 0.984519622445090559170399204414, 2.09748041020420391754801955811, 3.31600237701645880451615213888, 3.64355450074110639381330563881, 4.57111570898596251303621258598, 5.97129205032038572526985117408, 6.23297268385253889685720221186, 7.13298403690974437014487588763, 7.46223067263895439521490789935, 8.646363666160484527382602038718

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