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 + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s − 2.99·9-s + (−3 + 5.19i)11-s + (−3 − 5.19i)13-s + (−1.49 + 0.866i)15-s − 2·17-s − 7·19-s + (1.49 + 0.866i)21-s + (−0.5 − 0.866i)23-s + (2 − 3.46i)25-s − 5.19i·27-s + (−1 + 1.73i)29-s + (5 + 8.66i)31-s + ⋯
L(s)  = 1  + 0.999i·3-s + (0.223 + 0.387i)5-s + (0.188 − 0.327i)7-s − 0.999·9-s + (−0.904 + 1.56i)11-s + (−0.832 − 1.44i)13-s + (−0.387 + 0.223i)15-s − 0.485·17-s − 1.60·19-s + (0.327 + 0.188i)21-s + (−0.104 − 0.180i)23-s + (0.400 − 0.692i)25-s − 0.999i·27-s + (−0.185 + 0.321i)29-s + (0.898 + 1.55i)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  =  \((2,\ 1008,\ (\ :1/2),\ -0.939 + 0.342i)\)
\(L(1)\)  \(\approx\)  \(0.4810051584\)
\(L(\frac12)\)  \(\approx\)  \(0.4810051584\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (-4 - 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5 - 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4T + 89T^{2} \)
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

−10.41334741536988331336256046221, −9.975751279554894950698185171397, −8.796483284033356513921534811987, −8.019502367706033541894703541010, −7.10000505504824821251188717567, −6.10039384850856176607967336767, −4.79050826078438600298430538030, −4.67022628736874351729717458984, −3.11584046593279726622395732545, −2.27866574135177906016483310826, 0.19802837887515089557502944622, 1.89066272747593138913215682996, 2.68656046954670565157106998634, 4.20506971988925797770657569798, 5.38471656937120029026824900334, 6.08688729004508515663104924858, 6.94412296329149120012633734727, 7.890511484569878029122806067196, 8.709501614789002665773507504302, 9.117398423675225066619430229700

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