Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.888 + 0.458i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.27·5-s i·7-s + 6.57i·11-s − 4.05i·13-s + 6.09i·17-s + 2.08·19-s − 6.85·23-s − 3.37·25-s − 6.53·29-s − 3.26i·31-s − 1.27i·35-s − 2.95i·37-s − 3.35i·41-s − 10.9·43-s − 7.12·47-s + ⋯
L(s)  = 1  + 0.570·5-s − 0.377i·7-s + 1.98i·11-s − 1.12i·13-s + 1.47i·17-s + 0.477·19-s − 1.42·23-s − 0.674·25-s − 1.21·29-s − 0.587i·31-s − 0.215i·35-s − 0.485i·37-s − 0.523i·41-s − 1.67·43-s − 1.03·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.888 + 0.458i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.888 + 0.458i)$
$L(1)$  $\approx$  $0.2340378758$
$L(\frac12)$  $\approx$  $0.2340378758$
$L(\frac{3}{2})$   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 \( 1 - 1.27T + 5T^{2} \)
11 \( 1 - 6.57iT - 11T^{2} \)
13 \( 1 + 4.05iT - 13T^{2} \)
17 \( 1 - 6.09iT - 17T^{2} \)
19 \( 1 - 2.08T + 19T^{2} \)
23 \( 1 + 6.85T + 23T^{2} \)
29 \( 1 + 6.53T + 29T^{2} \)
31 \( 1 + 3.26iT - 31T^{2} \)
37 \( 1 + 2.95iT - 37T^{2} \)
41 \( 1 + 3.35iT - 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 + 7.12T + 47T^{2} \)
53 \( 1 - 2.87T + 53T^{2} \)
59 \( 1 + 7.75iT - 59T^{2} \)
61 \( 1 + 12.0iT - 61T^{2} \)
67 \( 1 - 3.01T + 67T^{2} \)
71 \( 1 + 3.48T + 71T^{2} \)
73 \( 1 - 2.76T + 73T^{2} \)
79 \( 1 - 0.849iT - 79T^{2} \)
83 \( 1 + 15.8iT - 83T^{2} \)
89 \( 1 - 4.06iT - 89T^{2} \)
97 \( 1 + 4.90T + 97T^{2} \)
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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

−7.83755869476044071818695762145, −7.13016519853756850565238782054, −6.30479723682129267892157052861, −5.66129414188164191424229524850, −4.93056333575656050571848224648, −4.05498794056853836389176744668, −3.43341463055469813888061937688, −1.96407207471704701164791342227, −1.82784888773182119498559785929, −0.05435972671418144870451048312, 1.28999644412899235549868757988, 2.27048124687890943092640491731, 3.14630783895056738216945727673, 3.85811643988854994364595112616, 4.93439595518967938900356390395, 5.62177974883195678078151498958, 6.14769322346767331279996349617, 6.84915715676254126630693127792, 7.72626883648052630579073553754, 8.474670654957056051338662944326

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