Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $0.605 + 0.795i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)5-s + (0.5 + 2.59i)7-s − 2·13-s + (3 − 5.19i)17-s + (0.5 + 0.866i)19-s + (−1 − 1.73i)23-s + (0.500 − 0.866i)25-s + 6·29-s + (−0.5 + 0.866i)31-s + (4 − 3.46i)35-s + (−1 − 1.73i)37-s + 2·41-s + 9·43-s + (1 + 1.73i)47-s + (−6.5 + 2.59i)49-s + ⋯
L(s)  = 1  + (−0.447 − 0.774i)5-s + (0.188 + 0.981i)7-s − 0.554·13-s + (0.727 − 1.26i)17-s + (0.114 + 0.198i)19-s + (−0.208 − 0.361i)23-s + (0.100 − 0.173i)25-s + 1.11·29-s + (−0.0898 + 0.155i)31-s + (0.676 − 0.585i)35-s + (−0.164 − 0.284i)37-s + 0.312·41-s + 1.37·43-s + (0.145 + 0.252i)47-s + (−0.928 + 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.605 + 0.795i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (865, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1512,\ (\ :1/2),\ 0.605 + 0.795i)\)
\(L(1)\)  \(\approx\)  \(1.446534722\)
\(L(\frac12)\)  \(\approx\)  \(1.446534722\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
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 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 9T + 43T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + (2.5 - 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (9 + 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - T + 97T^{2} \)
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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

−9.328452850962182473162756689369, −8.516394905834565590537238602637, −7.930409222946531544087263450389, −7.02293442072793526739936197948, −5.94503026095327874346981430824, −5.09134814278855467995815352418, −4.53062754939942534503311676295, −3.20222581647745350566525667192, −2.21503541101929305742060862499, −0.67869172730049211863501936009, 1.14381699843739766241159282098, 2.64211136845691535474448416617, 3.66675731695643390722182114447, 4.34679744044240182802171336437, 5.50816857353372588428262268864, 6.48800632249584503681377331316, 7.31684744172923089916144354756, 7.76348952180550052799532236895, 8.694665748065900063379766906093, 9.798295131946900679255190336462

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