Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $-0.482 + 0.875i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.667 + 0.667i)5-s − 7-s + (1.57 + 1.57i)11-s + (−1.83 + 1.83i)13-s − 3.40i·17-s + (−3.18 − 3.18i)19-s − 0.793i·23-s + 4.10i·25-s + (1.73 + 1.73i)29-s − 3.28i·31-s + (0.667 − 0.667i)35-s + (7.72 + 7.72i)37-s − 7.19·41-s + (5.84 − 5.84i)43-s − 13.0·47-s + ⋯
L(s)  = 1  + (−0.298 + 0.298i)5-s − 0.377·7-s + (0.475 + 0.475i)11-s + (−0.507 + 0.507i)13-s − 0.826i·17-s + (−0.730 − 0.730i)19-s − 0.165i·23-s + 0.821i·25-s + (0.322 + 0.322i)29-s − 0.589i·31-s + (0.112 − 0.112i)35-s + (1.26 + 1.26i)37-s − 1.12·41-s + (0.890 − 0.890i)43-s − 1.89·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.482 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.482 + 0.875i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (3599, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ -0.482 + 0.875i)$
$L(1)$  $\approx$  $0.5527757530$
$L(\frac12)$  $\approx$  $0.5527757530$
$L(\frac{3}{2})$   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 \( 1 + T \)
good5 \( 1 + (0.667 - 0.667i)T - 5iT^{2} \)
11 \( 1 + (-1.57 - 1.57i)T + 11iT^{2} \)
13 \( 1 + (1.83 - 1.83i)T - 13iT^{2} \)
17 \( 1 + 3.40iT - 17T^{2} \)
19 \( 1 + (3.18 + 3.18i)T + 19iT^{2} \)
23 \( 1 + 0.793iT - 23T^{2} \)
29 \( 1 + (-1.73 - 1.73i)T + 29iT^{2} \)
31 \( 1 + 3.28iT - 31T^{2} \)
37 \( 1 + (-7.72 - 7.72i)T + 37iT^{2} \)
41 \( 1 + 7.19T + 41T^{2} \)
43 \( 1 + (-5.84 + 5.84i)T - 43iT^{2} \)
47 \( 1 + 13.0T + 47T^{2} \)
53 \( 1 + (3.34 - 3.34i)T - 53iT^{2} \)
59 \( 1 + (7.41 + 7.41i)T + 59iT^{2} \)
61 \( 1 + (-1.93 + 1.93i)T - 61iT^{2} \)
67 \( 1 + (6.38 + 6.38i)T + 67iT^{2} \)
71 \( 1 + 3.41iT - 71T^{2} \)
73 \( 1 + 8.13iT - 73T^{2} \)
79 \( 1 - 0.0502iT - 79T^{2} \)
83 \( 1 + (-2.29 + 2.29i)T - 83iT^{2} \)
89 \( 1 - 7.18T + 89T^{2} \)
97 \( 1 - 1.49T + 97T^{2} \)
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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

−8.112330994087457485210208265220, −7.39516006130012596684621658911, −6.71533843712065122990432913081, −6.25188816240101973366356510280, −4.96864251526043380525572143510, −4.53675747592416587665766709656, −3.48406252571310439892343921992, −2.71353639014363494555076776164, −1.67705403016071029023679507008, −0.16838315010599786905743819494, 1.13197914613977022782358415453, 2.33684231727424979216937694257, 3.34570837707339974757119276809, 4.09910798833770205353140581637, 4.84219284667657110563024301297, 5.95449121230099641322412299861, 6.28033012775383808850722401680, 7.28511065806946209683227629496, 8.114509951688031167598923412618, 8.506917309614075134608788118292

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