Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.938 + 0.346i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.09·5-s + i·7-s − 0.646i·11-s − 4.37i·13-s − 0.295i·17-s + 5.29·19-s + 3.94·23-s + 4.58·25-s − 5.03·29-s + 9.16i·31-s + 3.09i·35-s − 2.55i·37-s − 10.4i·41-s + 1.63·43-s + 5.65·47-s + ⋯
L(s)  = 1  + 1.38·5-s + 0.377i·7-s − 0.194i·11-s − 1.21i·13-s − 0.0715i·17-s + 1.21·19-s + 0.823·23-s + 0.916·25-s − 0.934·29-s + 1.64i·31-s + 0.523i·35-s − 0.419i·37-s − 1.62i·41-s + 0.249·43-s + 0.825·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.346i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.938 + 0.346i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (2591, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.938 + 0.346i)$
$L(1)$  $\approx$  $2.672892561$
$L(\frac12)$  $\approx$  $2.672892561$
$L(\frac{3}{2})$   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 \( 1 - iT \)
good5 \( 1 - 3.09T + 5T^{2} \)
11 \( 1 + 0.646iT - 11T^{2} \)
13 \( 1 + 4.37iT - 13T^{2} \)
17 \( 1 + 0.295iT - 17T^{2} \)
19 \( 1 - 5.29T + 19T^{2} \)
23 \( 1 - 3.94T + 23T^{2} \)
29 \( 1 + 5.03T + 29T^{2} \)
31 \( 1 - 9.16iT - 31T^{2} \)
37 \( 1 + 2.55iT - 37T^{2} \)
41 \( 1 + 10.4iT - 41T^{2} \)
43 \( 1 - 1.63T + 43T^{2} \)
47 \( 1 - 5.65T + 47T^{2} \)
53 \( 1 - 8.64T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 6.92iT - 61T^{2} \)
67 \( 1 - 2.55T + 67T^{2} \)
71 \( 1 + 1.11T + 71T^{2} \)
73 \( 1 + 9.58T + 73T^{2} \)
79 \( 1 + 16.7iT - 79T^{2} \)
83 \( 1 - 8.50iT - 83T^{2} \)
89 \( 1 - 10.4iT - 89T^{2} \)
97 \( 1 + 2.41T + 97T^{2} \)
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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

−8.625268176607400406730932685583, −7.55222064647235483342162987196, −6.97973369079446454514295356664, −5.92793574849634871239310232295, −5.50079204180040841871972895416, −5.00605107300911761079948680600, −3.56770362408708980483863939893, −2.84646165518699784038249342948, −1.94838550390134719203087879337, −0.880122482303472240044464635947, 1.11144066092977623967537942955, 1.98020939971037556012887490198, 2.83616340736826752750846085432, 3.97015570602627701201767208045, 4.77042010680623732791264680160, 5.64253195822907093500451820381, 6.17069412830034127847430221163, 7.05344619921147804855960833102, 7.54968449734941311873315414754, 8.680694140592326464874584575136

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