Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $0.766 + 0.642i$
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)3-s + (−1 − 1.73i)5-s + (−0.5 + 0.866i)7-s + (1.5 + 2.59i)9-s + (0.5 − 0.866i)11-s + (3 + 5.19i)13-s + 3.46i·15-s − 5·17-s + 7·19-s + (1.5 − 0.866i)21-s + (2 + 3.46i)23-s + (0.500 − 0.866i)25-s − 5.19i·27-s + (2 − 3.46i)29-s + (−3 − 5.19i)31-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s + (−0.447 − 0.774i)5-s + (−0.188 + 0.327i)7-s + (0.5 + 0.866i)9-s + (0.150 − 0.261i)11-s + (0.832 + 1.44i)13-s + 0.894i·15-s − 1.21·17-s + 1.60·19-s + (0.327 − 0.188i)21-s + (0.417 + 0.722i)23-s + (0.100 − 0.173i)25-s − 0.999i·27-s + (0.371 − 0.643i)29-s + (−0.538 − 0.933i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.766 + 0.642i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (673, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ 0.766 + 0.642i)\)
\(L(1)\)  \(\approx\)  \(1.066941349\)
\(L(\frac12)\)  \(\approx\)  \(1.066941349\)
\(L(\frac{3}{2})\)   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 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3 - 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2 + 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (3.5 + 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + (3 - 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8 + 13.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.750602134832573164493971686400, −9.024756521375990362504658290706, −8.232515500873521451287137570030, −7.21114181375660063017265460879, −6.47773240772197719564003062171, −5.59414197443045563636132795290, −4.69645324057902840504579966857, −3.80716737012656742753774995804, −2.07452330142743886164965910853, −0.812235298555252510886035275990, 0.906756118738555286380305438060, 3.02128274132811354066849985770, 3.74294154192593674966189724795, 4.88781362742784228309991891096, 5.72133422824476750017118463218, 6.77451638415183494774830382539, 7.23215565870312661102432643279, 8.437375492676719228509422488476, 9.390380442065762142082684866181, 10.36888140826630746176488953348

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