Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $0.939 - 0.342i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·3-s + (1.5 + 2.59i)5-s + (0.5 − 0.866i)7-s − 2.99·9-s + (−3 + 5.19i)11-s + (−1 − 1.73i)13-s + (4.5 − 2.59i)15-s + 6·17-s + 7·19-s + (−1.49 − 0.866i)21-s + (1.5 + 2.59i)23-s + (−2 + 3.46i)25-s + 5.19i·27-s + (−3 + 5.19i)29-s + (1 + 1.73i)31-s + ⋯
L(s)  = 1  − 0.999i·3-s + (0.670 + 1.16i)5-s + (0.188 − 0.327i)7-s − 0.999·9-s + (−0.904 + 1.56i)11-s + (−0.277 − 0.480i)13-s + (1.16 − 0.670i)15-s + 1.45·17-s + 1.60·19-s + (−0.327 − 0.188i)21-s + (0.312 + 0.541i)23-s + (−0.400 + 0.692i)25-s + 0.999i·27-s + (−0.557 + 0.964i)29-s + (0.179 + 0.311i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.939 - 0.342i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (673, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ 0.939 - 0.342i)\)
\(L(1)\)  \(\approx\)  \(1.716254458\)
\(L(\frac12)\)  \(\approx\)  \(1.716254458\)
\(L(\frac{3}{2})\)   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 \( 1 + 1.73iT \)
7 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1 + 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (-2.5 + 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.10988741422847538334413575735, −9.383947140078458358520778779524, −7.903983053165542946697583603814, −7.36426437142942125919744457445, −6.97158293428436286731633916590, −5.69234507888719946771370170362, −5.15389003042644954330046899002, −3.30663373907994560975155351255, −2.55325418531545072069682681972, −1.39981180678212039421183646151, 0.872297971572014747466251686579, 2.63813215317811622380824500758, 3.63935879178098036372027016555, 4.94992987742234321206390062877, 5.43571162268061002545060068987, 6.01251393840904466698967811897, 7.79846340776504668548893833396, 8.377156827410472067792029340427, 9.307061164784269253296220964657, 9.673652374575824079255213528516

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