Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $0.921 - 0.388i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.26 + 2.19i)5-s + (2.64 + 0.160i)7-s + (1.19 − 2.06i)11-s − 0.461·13-s + (3.64 − 6.30i)17-s + (4.32 + 7.48i)19-s + (−1.49 − 2.58i)23-s + (−0.721 + 1.24i)25-s − 3.55·29-s + (4.15 − 7.18i)31-s + (2.99 + 6.00i)35-s + (1.10 + 1.90i)37-s − 6.81·41-s + 2.38·43-s + (4.21 + 7.30i)47-s + ⋯
L(s)  = 1  + (0.567 + 0.983i)5-s + (0.998 + 0.0607i)7-s + (0.358 − 0.621i)11-s − 0.128·13-s + (0.883 − 1.52i)17-s + (0.991 + 1.71i)19-s + (−0.310 − 0.538i)23-s + (−0.144 + 0.249i)25-s − 0.660·29-s + (0.745 − 1.29i)31-s + (0.506 + 1.01i)35-s + (0.181 + 0.313i)37-s − 1.06·41-s + 0.363·43-s + (0.615 + 1.06i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.921 - 0.388i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (865, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1512,\ (\ :1/2),\ 0.921 - 0.388i)\)
\(L(1)\)  \(\approx\)  \(2.264147774\)
\(L(\frac12)\)  \(\approx\)  \(2.264147774\)
\(L(\frac{3}{2})\)   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 \( 1 + (-2.64 - 0.160i)T \)
good5 \( 1 + (-1.26 - 2.19i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.19 + 2.06i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.461T + 13T^{2} \)
17 \( 1 + (-3.64 + 6.30i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.32 - 7.48i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.49 + 2.58i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.55T + 29T^{2} \)
31 \( 1 + (-4.15 + 7.18i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.10 - 1.90i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.81T + 41T^{2} \)
43 \( 1 - 2.38T + 43T^{2} \)
47 \( 1 + (-4.21 - 7.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.181 + 0.313i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.73 - 6.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.08 + 5.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.87 - 10.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.72T + 71T^{2} \)
73 \( 1 + (4.67 - 8.10i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.97 - 8.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.20T + 83T^{2} \)
89 \( 1 + (3.29 + 5.70i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.6T + 97T^{2} \)
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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

−9.773505419541202903378659874836, −8.696650965632960218695774913859, −7.75702409751209202439241167682, −7.29157686146219055529956203785, −6.07002394715772798637859778248, −5.62804309503546859199231984345, −4.50621734304248538862176138009, −3.34697309509325148818584609938, −2.47507232106565847224502883662, −1.22228501766281975872403110307, 1.17925045362973926136832937671, 1.92774144951291400983682395395, 3.45005141234438452511195762171, 4.65350690897521298776286836587, 5.12273883320387160833441648150, 5.98342791165406855587889206392, 7.14287600117997241855574439718, 7.85604512919962892688222819780, 8.777499960193607706694349142390, 9.246336874604424887151233983322

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