Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $-0.691 + 0.722i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.11·5-s + (−1.37 − 2.26i)7-s − 0.812·11-s + 2·13-s − 3.55i·17-s + 4.52i·19-s + 3.63i·23-s − 3.75·25-s − 8.95i·29-s − 5.08·31-s + (−1.53 − 2.52i)35-s − 0.718i·37-s − 10.4i·41-s + 1.11·43-s + 8.55·47-s + ⋯
L(s)  = 1  + 0.498·5-s + (−0.518 − 0.854i)7-s − 0.245·11-s + 0.554·13-s − 0.862i·17-s + 1.03i·19-s + 0.758i·23-s − 0.751·25-s − 1.66i·29-s − 0.914·31-s + (−0.258 − 0.426i)35-s − 0.118i·37-s − 1.63i·41-s + 0.170·43-s + 1.24·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.691 + 0.722i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (1567, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ -0.691 + 0.722i)$
$L(1)$  $\approx$  $1.056216052$
$L(\frac12)$  $\approx$  $1.056216052$
$L(\frac{3}{2})$   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 \( 1 + (1.37 + 2.26i)T \)
good5 \( 1 - 1.11T + 5T^{2} \)
11 \( 1 + 0.812T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 3.55iT - 17T^{2} \)
19 \( 1 - 4.52iT - 19T^{2} \)
23 \( 1 - 3.63iT - 23T^{2} \)
29 \( 1 + 8.95iT - 29T^{2} \)
31 \( 1 + 5.08T + 31T^{2} \)
37 \( 1 + 0.718iT - 37T^{2} \)
41 \( 1 + 10.4iT - 41T^{2} \)
43 \( 1 - 1.11T + 43T^{2} \)
47 \( 1 - 8.55T + 47T^{2} \)
53 \( 1 + 5.08iT - 53T^{2} \)
59 \( 1 + 2.23iT - 59T^{2} \)
61 \( 1 - 5.27T + 61T^{2} \)
67 \( 1 + 15.1T + 67T^{2} \)
71 \( 1 - 5.87iT - 71T^{2} \)
73 \( 1 + 3.86iT - 73T^{2} \)
79 \( 1 + 1.70iT - 79T^{2} \)
83 \( 1 + 7.27iT - 83T^{2} \)
89 \( 1 - 3.37iT - 89T^{2} \)
97 \( 1 - 8.55iT - 97T^{2} \)
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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

−7.978612439536502563548658395715, −7.50142518298431981738469837582, −6.72026272445872391627327363303, −5.85744783767649351633033221142, −5.41613643506916836045933398709, −4.12121611692954242805494930970, −3.69801660414401146521958575335, −2.57281981010476294184996595781, −1.56042300046404868427581920327, −0.29376694343137726575926845960, 1.37576482807414501065882195864, 2.41236765596003841162529901204, 3.14687698555668675139439737873, 4.14447033083242216963169350957, 5.13069543245349241813532731035, 5.82141408435978332016306628593, 6.39174352521593641358052342842, 7.14747554667080398991847376894, 8.114326530975171893694304586662, 8.877780034240087137927480830058

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