Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.912 - 0.408i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.17 − 1.17i)5-s − 7-s + (4.54 + 4.54i)11-s + (2.56 − 2.56i)13-s + 2.05i·17-s + (3.64 + 3.64i)19-s − 2.27i·23-s + 2.21i·25-s + (0.544 + 0.544i)29-s − 10.1i·31-s + (−1.17 + 1.17i)35-s + (4.71 + 4.71i)37-s − 0.487·41-s + (−7.56 + 7.56i)43-s − 0.768·47-s + ⋯
L(s)  = 1  + (0.527 − 0.527i)5-s − 0.377·7-s + (1.37 + 1.37i)11-s + (0.710 − 0.710i)13-s + 0.497i·17-s + (0.836 + 0.836i)19-s − 0.473i·23-s + 0.443i·25-s + (0.101 + 0.101i)29-s − 1.81i·31-s + (−0.199 + 0.199i)35-s + (0.775 + 0.775i)37-s − 0.0761·41-s + (−1.15 + 1.15i)43-s − 0.112·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.912 - 0.408i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (3599, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.912 - 0.408i)$
$L(1)$  $\approx$  $2.344005594$
$L(\frac12)$  $\approx$  $2.344005594$
$L(\frac{3}{2})$   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 \( 1 + T \)
good5 \( 1 + (-1.17 + 1.17i)T - 5iT^{2} \)
11 \( 1 + (-4.54 - 4.54i)T + 11iT^{2} \)
13 \( 1 + (-2.56 + 2.56i)T - 13iT^{2} \)
17 \( 1 - 2.05iT - 17T^{2} \)
19 \( 1 + (-3.64 - 3.64i)T + 19iT^{2} \)
23 \( 1 + 2.27iT - 23T^{2} \)
29 \( 1 + (-0.544 - 0.544i)T + 29iT^{2} \)
31 \( 1 + 10.1iT - 31T^{2} \)
37 \( 1 + (-4.71 - 4.71i)T + 37iT^{2} \)
41 \( 1 + 0.487T + 41T^{2} \)
43 \( 1 + (7.56 - 7.56i)T - 43iT^{2} \)
47 \( 1 + 0.768T + 47T^{2} \)
53 \( 1 + (0.269 - 0.269i)T - 53iT^{2} \)
59 \( 1 + (-0.0979 - 0.0979i)T + 59iT^{2} \)
61 \( 1 + (7.41 - 7.41i)T - 61iT^{2} \)
67 \( 1 + (6.83 + 6.83i)T + 67iT^{2} \)
71 \( 1 - 8.66iT - 71T^{2} \)
73 \( 1 - 13.6iT - 73T^{2} \)
79 \( 1 - 9.29iT - 79T^{2} \)
83 \( 1 + (-9.76 + 9.76i)T - 83iT^{2} \)
89 \( 1 + 7.27T + 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−8.499067260767964888682606264773, −7.81505738212527514229557604329, −6.96899659538756083319087601597, −6.17609325860565332364729361223, −5.69474307733801068296866884443, −4.63263952922408867963679944259, −3.99199215283738263733368394760, −3.06994851599456193902319706146, −1.80468552097206280763861978958, −1.12522779398748965560093956511, 0.790961129300136761849758157457, 1.84638104359926632899042507062, 3.14660438537510073585305322242, 3.49203845221481255304930644478, 4.61170613270201519842737779352, 5.58543024783913663800070372377, 6.39358771142690650755305628035, 6.65144049522077888680180866416, 7.54011433991089507467182799274, 8.708197884272695246052566482390

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