Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $0.187 + 0.982i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.119 + 0.207i)5-s + (0.710 − 2.54i)7-s + (1.21 − 2.09i)11-s + 0.760·13-s + (−1.71 + 2.96i)17-s + (−0.590 − 1.02i)19-s + (−1.09 − 1.88i)23-s + (2.47 − 4.28i)25-s + 2.89·29-s + (2.32 − 4.02i)31-s + (0.612 − 0.157i)35-s + (−2.89 − 5.00i)37-s − 8.54·41-s − 3.37·43-s + (2.58 + 4.47i)47-s + ⋯
L(s)  = 1  + (0.0534 + 0.0926i)5-s + (0.268 − 0.963i)7-s + (0.364 − 0.632i)11-s + 0.211·13-s + (−0.414 + 0.718i)17-s + (−0.135 − 0.234i)19-s + (−0.227 − 0.394i)23-s + (0.494 − 0.856i)25-s + 0.538·29-s + (0.417 − 0.722i)31-s + (0.103 − 0.0266i)35-s + (−0.475 − 0.823i)37-s − 1.33·41-s − 0.515·43-s + (0.376 + 0.652i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.187 + 0.982i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (865, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1512,\ (\ :1/2),\ 0.187 + 0.982i)\)
\(L(1)\)  \(\approx\)  \(1.571964027\)
\(L(\frac12)\)  \(\approx\)  \(1.571964027\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.710 + 2.54i)T \)
good5 \( 1 + (-0.119 - 0.207i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.21 + 2.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.760T + 13T^{2} \)
17 \( 1 + (1.71 - 2.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.590 + 1.02i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.09 + 1.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.89T + 29T^{2} \)
31 \( 1 + (-2.32 + 4.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.89 + 5.00i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.54T + 41T^{2} \)
43 \( 1 + 3.37T + 43T^{2} \)
47 \( 1 + (-2.58 - 4.47i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.56 - 2.70i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.11 + 5.39i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.681 + 1.18i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.03 + 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.02T + 71T^{2} \)
73 \( 1 + (-3.91 + 6.77i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.27 + 3.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 + (2.73 + 4.74i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−9.253948519312525161695660958416, −8.402743936248046716819850081763, −7.83564478263857366314274179981, −6.67489082005918416498432148806, −6.29942325097310981743748403356, −5.02034191653548382110640043740, −4.17431520608749698858982480075, −3.35336153208492440410585512588, −2.00147238675535366587906015662, −0.64806325990727449707328948672, 1.45519481291397627519341149704, 2.52776863876765964664147453265, 3.61843871458610222786406055279, 4.84275919853703934889760582857, 5.36101777214188702659672372067, 6.50436366105192128891316498523, 7.10915659276531881258923782753, 8.273181839343894117800494355353, 8.779493444749344888992243172370, 9.605945475663119389293475416362

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