Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $-0.0633 - 0.997i$
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 + (2.5 + 0.866i)7-s + (−4.5 + 2.59i)11-s + 6.92i·13-s + (−3 + 1.73i)17-s + (−1 + 1.73i)19-s + (−6 − 3.46i)23-s + (−1 − 1.73i)25-s + 9·29-s + (−0.5 − 0.866i)31-s + (3 + 3.46i)35-s + (1 − 1.73i)37-s + 3.46i·41-s − 3.46i·43-s + (5.5 + 4.33i)49-s + ⋯
L(s)  = 1  + (0.670 + 0.387i)5-s + (0.944 + 0.327i)7-s + (−1.35 + 0.783i)11-s + 1.92i·13-s + (−0.727 + 0.420i)17-s + (−0.229 + 0.397i)19-s + (−1.25 − 0.722i)23-s + (−0.200 − 0.346i)25-s + 1.67·29-s + (−0.0898 − 0.155i)31-s + (0.507 + 0.585i)35-s + (0.164 − 0.284i)37-s + 0.541i·41-s − 0.528i·43-s + (0.785 + 0.618i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.0633 - 0.997i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (271, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ -0.0633 - 0.997i)\)
\(L(1)\)  \(\approx\)  \(1.564373530\)
\(L(\frac12)\)  \(\approx\)  \(1.564373530\)
\(L(\frac{3}{2})\)   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 \( 1 + (-2.5 - 0.866i)T \)
good5 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.5 - 2.59i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.92iT - 13T^{2} \)
17 \( 1 + (3 - 1.73i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6 + 3.46i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6 + 3.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.92iT - 71T^{2} \)
73 \( 1 + (-6 + 3.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.5 + 0.866i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 15T + 83T^{2} \)
89 \( 1 + (-9 - 5.19i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.66iT - 97T^{2} \)
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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

−10.30137223448153356439117196703, −9.392456913647069536027902577335, −8.482494123929145211904405253515, −7.76943450381297701986946427752, −6.68284493660494082509932457183, −6.02551987270255980345549577264, −4.80945232585049918090712002554, −4.25638641070291280378823536336, −2.35382533684508749216753455626, −1.98003035227879247281892375450, 0.68409286285042670480214725991, 2.20468572041481814113360409137, 3.25317360801401223048450108988, 4.74839696643501832681308898618, 5.36117816755152438890553138862, 6.08832961564354635157744949681, 7.50530572768532093589279947603, 8.100030970195245716863865335897, 8.730587142109839978131006742470, 9.975045519953463060562262920260

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