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  − 0.665i·5-s + i·7-s + 2.07·11-s − 5.55·13-s − 2.16i·17-s + 4.49i·19-s − 4.28·23-s + 4.55·25-s − 1.41i·29-s − 6.61i·31-s + 0.665·35-s + 5.43·37-s − 5.69i·41-s − 2.11i·43-s + 10.5·47-s + ⋯
L(s)  = 1  − 0.297i·5-s + 0.377i·7-s + 0.627·11-s − 1.54·13-s − 0.524i·17-s + 1.03i·19-s − 0.892·23-s + 0.911·25-s − 0.262i·29-s − 1.18i·31-s + 0.112·35-s + 0.894·37-s − 0.889i·41-s − 0.322i·43-s + 1.53·47-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  =  $(2,\ 4032,\ (\ :1/2),\ 0.816 + 0.577i)$
$L(1)$  $\approx$  $1.636533414$
$L(\frac12)$  $\approx$  $1.636533414$
$L(\frac{3}{2})$   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 \( 1 + 0.665iT - 5T^{2} \)
11 \( 1 - 2.07T + 11T^{2} \)
13 \( 1 + 5.55T + 13T^{2} \)
17 \( 1 + 2.16iT - 17T^{2} \)
19 \( 1 - 4.49iT - 19T^{2} \)
23 \( 1 + 4.28T + 23T^{2} \)
29 \( 1 + 1.41iT - 29T^{2} \)
31 \( 1 + 6.61iT - 31T^{2} \)
37 \( 1 - 5.43T + 37T^{2} \)
41 \( 1 + 5.69iT - 41T^{2} \)
43 \( 1 + 2.11iT - 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 - 10.6iT - 53T^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 - 0.615T + 61T^{2} \)
67 \( 1 + 14.0iT - 67T^{2} \)
71 \( 1 - 15.5T + 71T^{2} \)
73 \( 1 + 11.9T + 73T^{2} \)
79 \( 1 - 0.824iT - 79T^{2} \)
83 \( 1 + 6.36T + 83T^{2} \)
89 \( 1 - 12.6iT - 89T^{2} \)
97 \( 1 - 0.824T + 97T^{2} \)
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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

−8.360060573773636413830536576807, −7.61235166304192396688881058173, −7.02722927791593856247181580061, −6.03433489787495981011384294550, −5.46341489539800674243376720020, −4.54590480520091139118770791005, −3.90328083744405516378406358068, −2.70455296958549551120806528462, −1.98933741999365350588236977846, −0.60285681886471432230095864836, 0.875278314986834915823927579903, 2.17784153476863363437713306591, 2.97189243091398691759292514096, 3.99428340275457118491062983248, 4.71472347883789054495907779576, 5.46924620279390791921142056301, 6.56150049596875855719387815083, 6.96575300745726664277794480518, 7.67811400812115790353488572052, 8.547200039390650689585584996179

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