Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.482 + 0.875i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.381·5-s + i·7-s − 2.13i·11-s + 6.49i·13-s + 6.97i·17-s − 6.19·19-s − 1.82·23-s − 4.85·25-s − 0.694·29-s − 1.67i·31-s − 0.381i·35-s − 8.06i·37-s − 1.31i·41-s − 7.67·43-s + 6.82·47-s + ⋯
L(s)  = 1  − 0.170·5-s + 0.377i·7-s − 0.643i·11-s + 1.80i·13-s + 1.69i·17-s − 1.42·19-s − 0.380·23-s − 0.970·25-s − 0.129·29-s − 0.301i·31-s − 0.0644i·35-s − 1.32i·37-s − 0.204i·41-s − 1.17·43-s + 0.995·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.482 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.482 + 0.875i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.482 + 0.875i)$
$L(1)$  $\approx$  $0.3254716192$
$L(\frac12)$  $\approx$  $0.3254716192$
$L(\frac{3}{2})$   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 \( 1 + 0.381T + 5T^{2} \)
11 \( 1 + 2.13iT - 11T^{2} \)
13 \( 1 - 6.49iT - 13T^{2} \)
17 \( 1 - 6.97iT - 17T^{2} \)
19 \( 1 + 6.19T + 19T^{2} \)
23 \( 1 + 1.82T + 23T^{2} \)
29 \( 1 + 0.694T + 29T^{2} \)
31 \( 1 + 1.67iT - 31T^{2} \)
37 \( 1 + 8.06iT - 37T^{2} \)
41 \( 1 + 1.31iT - 41T^{2} \)
43 \( 1 + 7.67T + 43T^{2} \)
47 \( 1 - 6.82T + 47T^{2} \)
53 \( 1 - 0.954T + 53T^{2} \)
59 \( 1 + 12.8iT - 59T^{2} \)
61 \( 1 + 12.4iT - 61T^{2} \)
67 \( 1 + 0.634T + 67T^{2} \)
71 \( 1 - 5.79T + 71T^{2} \)
73 \( 1 - 8.13T + 73T^{2} \)
79 \( 1 + 14.1iT - 79T^{2} \)
83 \( 1 - 3.36iT - 83T^{2} \)
89 \( 1 - 15.9iT - 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−8.093514566469523719991890622201, −7.02253522807383789021281349043, −6.28832574371740939064522204136, −5.96646901245524098469143168681, −4.88666111096454457099713174590, −3.95079492138542271230321181503, −3.71217089031242948133520706833, −2.14957882588864389608731473706, −1.82939759112669168863148186484, −0.089036021360895998987872826137, 0.992862875220433885757262665387, 2.29704557327303454797600487326, 3.01873680010343787186163192561, 3.94335602495095979940465802277, 4.71915340432445668840027366022, 5.38577621038828299020065908761, 6.18568061182603640450932029334, 7.03515590620972252069266053190, 7.58290739918991035864162871688, 8.215167158658897391728773353267

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