Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.707 - 0.707i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.14i·5-s + i·7-s + 2.04·11-s + 2.44·13-s + 3.14i·17-s + 4.44i·19-s + 2.82·23-s − 4.89·25-s + 0.778i·29-s + 0.449i·31-s − 3.14·35-s − 7·37-s + 2.51i·41-s + 4.34i·43-s + 7.38·47-s + ⋯
L(s)  = 1  + 1.40i·5-s + 0.377i·7-s + 0.618·11-s + 0.679·13-s + 0.763i·17-s + 1.02i·19-s + 0.589·23-s − 0.979·25-s + 0.144i·29-s + 0.0807i·31-s − 0.531·35-s − 1.15·37-s + 0.392i·41-s + 0.663i·43-s + 1.07·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.707 - 0.707i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (2591, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.707 - 0.707i)$
$L(1)$  $\approx$  $1.931235003$
$L(\frac12)$  $\approx$  $1.931235003$
$L(\frac{3}{2})$   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 \( 1 - 3.14iT - 5T^{2} \)
11 \( 1 - 2.04T + 11T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
17 \( 1 - 3.14iT - 17T^{2} \)
19 \( 1 - 4.44iT - 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 - 0.778iT - 29T^{2} \)
31 \( 1 - 0.449iT - 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 - 2.51iT - 41T^{2} \)
43 \( 1 - 4.34iT - 43T^{2} \)
47 \( 1 - 7.38T + 47T^{2} \)
53 \( 1 - 6.29iT - 53T^{2} \)
59 \( 1 - 5.83T + 59T^{2} \)
61 \( 1 + 5.34T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 12.8T + 73T^{2} \)
79 \( 1 + 13.4iT - 79T^{2} \)
83 \( 1 - 6.75T + 83T^{2} \)
89 \( 1 + 14.1iT - 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−8.397082085384362547733609054713, −7.47371968536023046365922874642, −6.93783244649565450184391588844, −6.10859546427307442350465165493, −5.86923660362301260510379645309, −4.65230653866770991886957885538, −3.66943281249809802027528379249, −3.26167052563177931259275649550, −2.25641014701022329842338303078, −1.33666832799035833835007887596, 0.54380689866302118512548216040, 1.22124955038903024331015413269, 2.32560024535311574376018493353, 3.53591086787433434216554475521, 4.17046117564592244188250991410, 5.04686570271960202962683837518, 5.37155804526808275238023989559, 6.52304840510245321695218111674, 7.02428608087414218370263609475, 7.933133866406348666958008813935

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