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  =  $(2,\ 4032,\ (\ :1/2),\ -0.816 + 0.577i)$
$L(1)$  $\approx$  $0.06798823916$
$L(\frac12)$  $\approx$  $0.06798823916$
$L(\frac{3}{2})$   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

−7.952892309007474536074179865659, −7.58949103989693602000561800764, −6.65549157382613034555779863073, −5.96370931804924973030115093241, −5.07429278046845897150304361858, −4.52969185698805292843167467780, −3.29146752209230988894789142770, −2.64640438278198415028782047245, −1.66299146394683237871553114404, −0.01972896319906616089865538733, 1.19015302654336680262086316331, 2.60200613790308956010116932702, 3.04920680190547790969977675558, 4.56793232711051310392813215302, 4.78692656418839317553794671906, 5.62528530036187098080300311171, 6.78419297108960153951585830552, 7.24155622120168685525991579339, 7.88053172783892856455640768320, 8.872604953408134661196709086890

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