Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $-0.816 - 0.577i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.31i·5-s i·7-s − 4.72·11-s + 4.97·13-s + 0.484i·17-s + 2.29i·19-s − 7.97·23-s − 5.97·25-s − 1.41i·29-s − 7.66i·31-s − 3.31·35-s + 2.39·37-s + 6.55i·41-s − 5.37i·43-s − 6.21·47-s + ⋯
L(s)  = 1  − 1.48i·5-s − 0.377i·7-s − 1.42·11-s + 1.38·13-s + 0.117i·17-s + 0.525i·19-s − 1.66·23-s − 1.19·25-s − 0.262i·29-s − 1.37i·31-s − 0.560·35-s + 0.393·37-s + 1.02i·41-s − 0.819i·43-s − 0.906·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.816 - 0.577i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (575, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ -0.816 - 0.577i)$
$L(1)$  $\approx$  $0.3954219695$
$L(\frac12)$  $\approx$  $0.3954219695$
$L(\frac{3}{2})$   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 \( 1 + 3.31iT - 5T^{2} \)
11 \( 1 + 4.72T + 11T^{2} \)
13 \( 1 - 4.97T + 13T^{2} \)
17 \( 1 - 0.484iT - 17T^{2} \)
19 \( 1 - 2.29iT - 19T^{2} \)
23 \( 1 + 7.97T + 23T^{2} \)
29 \( 1 + 1.41iT - 29T^{2} \)
31 \( 1 + 7.66iT - 31T^{2} \)
37 \( 1 - 2.39T + 37T^{2} \)
41 \( 1 - 6.55iT - 41T^{2} \)
43 \( 1 + 5.37iT - 43T^{2} \)
47 \( 1 + 6.21T + 47T^{2} \)
53 \( 1 - 1.00iT - 53T^{2} \)
59 \( 1 - 1.38T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 + 3.27iT - 67T^{2} \)
71 \( 1 + 3.34T + 71T^{2} \)
73 \( 1 + 2.10T + 73T^{2} \)
79 \( 1 + 12.0iT - 79T^{2} \)
83 \( 1 + 3.24T + 83T^{2} \)
89 \( 1 - 5.72iT - 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−7.946248228624446998836589217979, −7.73379031910990556241205801914, −6.20965286112737230381058334422, −5.84465115877036228946681210502, −4.94166135517582561639293686882, −4.26454159560204237363616811196, −3.51276252754482944860719173840, −2.19296232636556929616015510186, −1.23606982104351875331330739030, −0.11235887295961448298586996020, 1.74357723956476554793062401648, 2.76138026143399388495547430594, 3.23439284884824747630822249099, 4.23409663997593550323662894751, 5.33772035957502628886164385392, 6.02041743074750139597843706063, 6.63620241855020103985671122527, 7.41068811804609919593663050297, 8.107299099331134405279938160259, 8.715202757262024828889148138560

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