Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $0.991 + 0.126i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 0.866i)5-s + 2.64·7-s + (3.96 − 2.29i)11-s − 3.46i·13-s + (−4.5 + 2.59i)17-s + (1.32 − 2.29i)19-s + (3.96 + 2.29i)23-s + (−1 − 1.73i)25-s + (−1.32 − 2.29i)31-s + (3.96 + 2.29i)35-s + (−3.5 + 6.06i)37-s − 3.46i·41-s + 9.16i·43-s + (3.96 − 6.87i)47-s + 7.00·49-s + ⋯
L(s)  = 1  + (0.670 + 0.387i)5-s + 0.999·7-s + (1.19 − 0.690i)11-s − 0.960i·13-s + (−1.09 + 0.630i)17-s + (0.303 − 0.525i)19-s + (0.827 + 0.477i)23-s + (−0.200 − 0.346i)25-s + (−0.237 − 0.411i)31-s + (0.670 + 0.387i)35-s + (−0.575 + 0.996i)37-s − 0.541i·41-s + 1.39i·43-s + (0.578 − 1.00i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.991 + 0.126i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (271, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ 0.991 + 0.126i)\)
\(L(1)\)  \(\approx\)  \(2.115665273\)
\(L(\frac12)\)  \(\approx\)  \(2.115665273\)
\(L(\frac{3}{2})\)   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 \( 1 - 2.64T \)
good5 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.96 + 2.29i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (4.5 - 2.59i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.32 + 2.29i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.96 - 2.29i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (1.32 + 2.29i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 - 9.16iT - 43T^{2} \)
47 \( 1 + (-3.96 + 6.87i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.96 - 6.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 0.866i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.96 + 2.29i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.16iT - 71T^{2} \)
73 \( 1 + (4.5 - 2.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.96 + 2.29i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (1.5 + 0.866i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.46iT - 97T^{2} \)
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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

−10.00966489573113557064137425659, −8.989646640230076948410052316194, −8.459174659670076817924752917333, −7.40219822223365340816047759884, −6.47662414254567900730104175892, −5.72377639599791445657553349901, −4.74605152955831451290131254109, −3.64106911829651362794749256939, −2.42433849975433527769903920059, −1.20590323516215075444832225952, 1.40949386715488600927214932143, 2.20974371425420272304315663410, 3.95333522225174300825059342518, 4.71846521011306307977979935820, 5.56603854341519971068724878277, 6.74439535691993880593196688754, 7.27556897760004803016544550522, 8.615683952140882336754887959745, 9.100523809772954772429508176414, 9.783168150100075503798626046598

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