Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.902 - 0.430i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.76·5-s + (0.480 + 2.60i)7-s + 5.75·11-s + 2·13-s + 6.71i·17-s − 5.20i·19-s − 4.43i·23-s + 2.66·25-s − 1.54i·29-s + 8.05·31-s + (1.33 + 7.20i)35-s − 4.42i·37-s − 0.209i·41-s + 10.5·43-s − 4.58·47-s + ⋯
L(s)  = 1  + 1.23·5-s + (0.181 + 0.983i)7-s + 1.73·11-s + 0.554·13-s + 1.62i·17-s − 1.19i·19-s − 0.924i·23-s + 0.533·25-s − 0.286i·29-s + 1.44·31-s + (0.225 + 1.21i)35-s − 0.727i·37-s − 0.0326i·41-s + 1.60·43-s − 0.669·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.902 - 0.430i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (1567, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.902 - 0.430i)$
$L(1)$  $\approx$  $3.055323555$
$L(\frac12)$  $\approx$  $3.055323555$
$L(\frac{3}{2})$   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 \( 1 + (-0.480 - 2.60i)T \)
good5 \( 1 - 2.76T + 5T^{2} \)
11 \( 1 - 5.75T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 6.71iT - 17T^{2} \)
19 \( 1 + 5.20iT - 19T^{2} \)
23 \( 1 + 4.43iT - 23T^{2} \)
29 \( 1 + 1.54iT - 29T^{2} \)
31 \( 1 - 8.05T + 31T^{2} \)
37 \( 1 + 4.42iT - 37T^{2} \)
41 \( 1 + 0.209iT - 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 + 4.58T + 47T^{2} \)
53 \( 1 - 8.05iT - 53T^{2} \)
59 \( 1 + 5.53iT - 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 + 4.04T + 67T^{2} \)
71 \( 1 - 1.10iT - 71T^{2} \)
73 \( 1 + 9.59iT - 73T^{2} \)
79 \( 1 + 14.7iT - 79T^{2} \)
83 \( 1 - 8.86iT - 83T^{2} \)
89 \( 1 - 13.6iT - 89T^{2} \)
97 \( 1 + 4.58iT - 97T^{2} \)
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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

−8.768920318675575934150550537148, −7.927387467367870148362239123379, −6.63954204640471995706439672570, −6.21931281894246841888916432648, −5.84040039649677392092356564294, −4.71974946907942806512336193211, −3.99824513835309935930150172707, −2.82147308109784252436732539847, −1.98667923812429529081835840817, −1.19369671947814296886386512867, 1.09008592049863520992888098206, 1.59936073333356926646443002196, 2.91232478097438326169409927914, 3.83440025253934210447415831696, 4.55979139775842548227559752616, 5.52992437025702896528866324126, 6.25294193943978628805137485050, 6.80602281594299158873660700170, 7.55793862395211532620706616148, 8.463034884463876510790450798828

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