Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $-0.912 - 0.409i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.08i·5-s + (−1.08 + 2.41i)7-s + 2i·11-s + 4.14i·13-s + 7.39i·17-s + 4.77·19-s + 3.65i·23-s + 3.82·25-s − 7.65·29-s + 7.39·31-s + (−2.61 − 1.17i)35-s − 3.65·37-s − 8.28i·41-s − 7.65i·43-s + 3.06·47-s + ⋯
L(s)  = 1  + 0.484i·5-s + (−0.409 + 0.912i)7-s + 0.603i·11-s + 1.14i·13-s + 1.79i·17-s + 1.09·19-s + 0.762i·23-s + 0.765·25-s − 1.42·29-s + 1.32·31-s + (−0.441 − 0.198i)35-s − 0.601·37-s − 1.29i·41-s − 1.16i·43-s + 0.446·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.912 - 0.409i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (3583, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ -0.912 - 0.409i)$
$L(1)$  $\approx$  $1.426162014$
$L(\frac12)$  $\approx$  $1.426162014$
$L(\frac{3}{2})$   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 \( 1 + (1.08 - 2.41i)T \)
good5 \( 1 - 1.08iT - 5T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 - 4.14iT - 13T^{2} \)
17 \( 1 - 7.39iT - 17T^{2} \)
19 \( 1 - 4.77T + 19T^{2} \)
23 \( 1 - 3.65iT - 23T^{2} \)
29 \( 1 + 7.65T + 29T^{2} \)
31 \( 1 - 7.39T + 31T^{2} \)
37 \( 1 + 3.65T + 37T^{2} \)
41 \( 1 + 8.28iT - 41T^{2} \)
43 \( 1 + 7.65iT - 43T^{2} \)
47 \( 1 - 3.06T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 5.67T + 59T^{2} \)
61 \( 1 - 1.08iT - 61T^{2} \)
67 \( 1 + 4.34iT - 67T^{2} \)
71 \( 1 - 3.17iT - 71T^{2} \)
73 \( 1 + 0.896iT - 73T^{2} \)
79 \( 1 - 7.17iT - 79T^{2} \)
83 \( 1 + 1.71T + 83T^{2} \)
89 \( 1 - 5.22iT - 89T^{2} \)
97 \( 1 - 11.7iT - 97T^{2} \)
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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

−8.906944398215207429753929572703, −8.029288849296785640335515285898, −7.17549904053380495537172193613, −6.60799093616617781548821190140, −5.82580594497085289396968790329, −5.16486359338859866072902675197, −4.04974008065376233723139258344, −3.40265303953789240958207852430, −2.32487986298231511032584345175, −1.58136203424289779083866351139, 0.45146470965578731625101749957, 1.14233314700413731101887736021, 2.89690700382986480799362859085, 3.21977941674041807098400259881, 4.47126990863056720593987061709, 5.04244679481903873371170885217, 5.86732318403573258487695097003, 6.72976187057320564883322811130, 7.48993106658769268361443857300, 7.992244964938907919274787412273

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