Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.938 + 0.346i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.646·5-s + i·7-s + 3.09i·11-s + 0.913i·13-s − 6.77i·17-s − 5.29·19-s − 2.53·23-s − 4.58·25-s + 9.93·29-s − 9.16i·31-s − 0.646i·35-s − 7.84i·37-s + 9.01i·41-s + 12.2·43-s + 5.65·47-s + ⋯
L(s)  = 1  − 0.288·5-s + 0.377i·7-s + 0.933i·11-s + 0.253i·13-s − 1.64i·17-s − 1.21·19-s − 0.528·23-s − 0.916·25-s + 1.84·29-s − 1.64i·31-s − 0.109i·35-s − 1.28i·37-s + 1.40i·41-s + 1.86·43-s + 0.825·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.346i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.938 + 0.346i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (2591, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.938 + 0.346i)$
$L(1)$  $\approx$  $1.507127186$
$L(\frac12)$  $\approx$  $1.507127186$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 + 0.646T + 5T^{2} \)
11 \( 1 - 3.09iT - 11T^{2} \)
13 \( 1 - 0.913iT - 13T^{2} \)
17 \( 1 + 6.77iT - 17T^{2} \)
19 \( 1 + 5.29T + 19T^{2} \)
23 \( 1 + 2.53T + 23T^{2} \)
29 \( 1 - 9.93T + 29T^{2} \)
31 \( 1 + 9.16iT - 31T^{2} \)
37 \( 1 + 7.84iT - 37T^{2} \)
41 \( 1 - 9.01iT - 41T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 - 5.65T + 47T^{2} \)
53 \( 1 - 1.15T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 6.92iT - 61T^{2} \)
67 \( 1 - 7.84T + 67T^{2} \)
71 \( 1 - 5.36T + 71T^{2} \)
73 \( 1 + 0.417T + 73T^{2} \)
79 \( 1 - 10.7iT - 79T^{2} \)
83 \( 1 - 15.9iT - 83T^{2} \)
89 \( 1 + 9.01iT - 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−8.269718339952688233470327608716, −7.72814263323340807146586914336, −6.95570712915239493604691294538, −6.25182774301847625845359576642, −5.41297959514633919737207275293, −4.45880429627601369538802227373, −4.05724690543260640475701009415, −2.64445839925765612367423455824, −2.16157982620422420272345500687, −0.58883657103295071608592768301, 0.816363954498810292737362315756, 1.99713703496434975495584851528, 3.12421944071401971005400486475, 3.92857325406980701901552374691, 4.53127513792266685024996674021, 5.70023631736672270368403524011, 6.22419387014833846617187556300, 6.96339741013036181580944582763, 7.953083026104907420236843749133, 8.439606709565785461054617027748

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