Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.258 - 0.965i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.732i·5-s − 7-s − 1.46i·11-s + 3.26i·13-s − 2·17-s + 4.73i·19-s + 3.46·23-s + 4.46·25-s − 5.46i·29-s − 4·31-s + 0.732i·35-s − 5.46i·37-s + 2·41-s + 1.46i·43-s − 10.9·47-s + ⋯
L(s)  = 1  − 0.327i·5-s − 0.377·7-s − 0.441i·11-s + 0.906i·13-s − 0.485·17-s + 1.08i·19-s + 0.722·23-s + 0.892·25-s − 1.01i·29-s − 0.718·31-s + 0.123i·35-s − 0.898i·37-s + 0.312·41-s + 0.223i·43-s − 1.59·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.258 - 0.965i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (2017, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.258 - 0.965i)$
$L(1)$  $\approx$  $1.279569950$
$L(\frac12)$  $\approx$  $1.279569950$
$L(\frac{3}{2})$   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 \( 1 + T \)
good5 \( 1 + 0.732iT - 5T^{2} \)
11 \( 1 + 1.46iT - 11T^{2} \)
13 \( 1 - 3.26iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 4.73iT - 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + 5.46iT - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 5.46iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 1.46iT - 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 - 7.66iT - 59T^{2} \)
61 \( 1 - 13.1iT - 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 + 0.928T + 73T^{2} \)
79 \( 1 + 2.92T + 79T^{2} \)
83 \( 1 - 11.6iT - 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 + 4.92T + 97T^{2} \)
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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

−8.844960581044570451190111332249, −7.82586295271407634607811794561, −7.16227701957063019762158053507, −6.30439962037175868079926733380, −5.76350657422559029789120680018, −4.75226825423893273235075553954, −4.06816711069393023174207906793, −3.18347499421894553522735376704, −2.16341679187750412422639067135, −1.06119790628535093346913823740, 0.40776064959521645545769169560, 1.79502251677854224753786018614, 2.94947552660839131081949625188, 3.39314592152252274967408894875, 4.72753801948510179414020189152, 5.12068952021409522697721662617, 6.22892932815375096004856077072, 6.89085266550901358105328698790, 7.37205913130793129631039033610, 8.401660907060080285884545494943

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