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  =  $(2,\ 6048,\ (\ :1/2),\ -0.707 + 0.707i)$
$L(1)$  $\approx$  $0.03442758122$
$L(\frac12)$  $\approx$  $0.03442758122$
$L(\frac{3}{2})$   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

−7.76173701607546856814078215970, −6.99626469508748327688592224271, −6.52496324582115304427700126135, −5.83387853673663178296426297709, −4.92497757022656492821225302072, −3.96609296999329002912865494468, −3.28532678419523672433887537579, −2.57881324194787748312886762594, −1.58037579211822471829780583338, −0.008677174784077116089984647120, 1.20554259665473607627727570593, 2.03017784336611726609759652918, 3.17686898089117780141469024172, 4.03060166238075920396756836548, 4.87320404865296229692666548547, 5.38875000476502379798603979293, 6.03494979251750237689019184688, 6.91952524226457725158985374480, 8.030869966620906025866996844101, 8.172714548591770328806313699365

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