Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.755 + 0.654i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.85·5-s i·7-s + 4.98i·11-s + 2.35i·13-s + 8.19i·17-s − 5.90·19-s + 4.43·23-s + 3.17·25-s − 6.41·29-s + 3.31i·31-s + 2.85i·35-s − 5.51i·37-s − 3.41i·41-s − 1.37·43-s + 2.60·47-s + ⋯
L(s)  = 1  − 1.27·5-s − 0.377i·7-s + 1.50i·11-s + 0.652i·13-s + 1.98i·17-s − 1.35·19-s + 0.925·23-s + 0.634·25-s − 1.19·29-s + 0.595i·31-s + 0.483i·35-s − 0.905i·37-s − 0.533i·41-s − 0.209·43-s + 0.379·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.755 + 0.654i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.755 + 0.654i)$
$L(1)$  $\approx$  $0.1944309066$
$L(\frac12)$  $\approx$  $0.1944309066$
$L(\frac{3}{2})$   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 \( 1 + 2.85T + 5T^{2} \)
11 \( 1 - 4.98iT - 11T^{2} \)
13 \( 1 - 2.35iT - 13T^{2} \)
17 \( 1 - 8.19iT - 17T^{2} \)
19 \( 1 + 5.90T + 19T^{2} \)
23 \( 1 - 4.43T + 23T^{2} \)
29 \( 1 + 6.41T + 29T^{2} \)
31 \( 1 - 3.31iT - 31T^{2} \)
37 \( 1 + 5.51iT - 37T^{2} \)
41 \( 1 + 3.41iT - 41T^{2} \)
43 \( 1 + 1.37T + 43T^{2} \)
47 \( 1 - 2.60T + 47T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
59 \( 1 - 10.6iT - 59T^{2} \)
61 \( 1 + 1.41iT - 61T^{2} \)
67 \( 1 + 0.221T + 67T^{2} \)
71 \( 1 + 0.398T + 71T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 - 9.76iT - 79T^{2} \)
83 \( 1 + 2.06iT - 83T^{2} \)
89 \( 1 + 17.6iT - 89T^{2} \)
97 \( 1 + 3.62T + 97T^{2} \)
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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

−8.678627168588501357666132001511, −7.55397882988649379421343754495, −7.29740867865866514904159042472, −6.57887126177945967694135612727, −5.67606885057738842299548438619, −4.58898835098931235974398076772, −4.08155501699619583129260542966, −3.70182752425059965333430727863, −2.30568565568410475465715517028, −1.51413382529173861148662732682, 0.06511277867035971586365332615, 0.835501587957087443117857669692, 2.48189162973743229319333360666, 3.17587741669727444361252707849, 3.83685221568063036428738973076, 4.75962647481907952134318578939, 5.43016008864169350920372327280, 6.22774314608448062370651914895, 7.07243390916032594610119010528, 7.68596626046361310783606097485

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