Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $-0.924 - 0.380i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.88·5-s + (−2.62 − 0.339i)7-s + 1.48·11-s + 2·13-s − 3.76i·17-s + 0.679i·19-s − 5.20i·23-s + 10.0·25-s − 7.03i·29-s + 6.42·31-s + (10.1 + 1.32i)35-s + 8.71i·37-s + 3.16i·41-s + 8.20·43-s − 9.88·47-s + ⋯
L(s)  = 1  − 1.73·5-s + (−0.991 − 0.128i)7-s + 0.446·11-s + 0.554·13-s − 0.913i·17-s + 0.155i·19-s − 1.08i·23-s + 2.01·25-s − 1.30i·29-s + 1.15·31-s + (1.72 + 0.223i)35-s + 1.43i·37-s + 0.493i·41-s + 1.25·43-s − 1.44·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.924 - 0.380i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (1567, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ -0.924 - 0.380i)$
$L(1)$  $\approx$  $0.02178647919$
$L(\frac12)$  $\approx$  $0.02178647919$
$L(\frac{3}{2})$   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 \( 1 + (2.62 + 0.339i)T \)
good5 \( 1 + 3.88T + 5T^{2} \)
11 \( 1 - 1.48T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 3.76iT - 17T^{2} \)
19 \( 1 - 0.679iT - 19T^{2} \)
23 \( 1 + 5.20iT - 23T^{2} \)
29 \( 1 + 7.03iT - 29T^{2} \)
31 \( 1 - 6.42T + 31T^{2} \)
37 \( 1 - 8.71iT - 37T^{2} \)
41 \( 1 - 3.16iT - 41T^{2} \)
43 \( 1 - 8.20T + 43T^{2} \)
47 \( 1 + 9.88T + 47T^{2} \)
53 \( 1 - 6.42iT - 53T^{2} \)
59 \( 1 - 7.76iT - 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 + 8.81T + 67T^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 + 13.4iT - 73T^{2} \)
79 \( 1 - 4.44iT - 79T^{2} \)
83 \( 1 - 10.4iT - 83T^{2} \)
89 \( 1 + 10.6iT - 89T^{2} \)
97 \( 1 + 9.88iT - 97T^{2} \)
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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

−7.963829428947653981060400361417, −7.40747497620712218202576729455, −6.54354846834221362658807253593, −6.10491928057944240126850024439, −4.58499503913830490280400565479, −4.33101816564393559303695480387, −3.30206239418491145113301103494, −2.81170078779686062408555210434, −0.982930203167179147124353260843, −0.008426563678264835924467125543, 1.27083299168847145997356186623, 2.83340995355514430512361160642, 3.72557415795863154847411659594, 3.91441093015374427705387410004, 5.04156381725785124434605020943, 6.04283344151542817904053195886, 6.75241104461477051687020954227, 7.41063587141254230967440910765, 8.100582570307051708259327459117, 8.796498695137236545107578536386

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