Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $-0.239 - 0.970i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0144 − 0.0250i)5-s + (−1.70 − 2.02i)7-s + (1.88 + 3.25i)11-s − 2.97·13-s + (−0.708 − 1.22i)17-s + (−3.81 + 6.60i)19-s + (2.98 − 5.17i)23-s + (2.49 + 4.32i)25-s − 9.99·29-s + (4.27 + 7.40i)31-s + (−0.0753 + 0.0135i)35-s + (−0.737 + 1.27i)37-s + 4.38·41-s + 3.76·43-s + (−2.07 + 3.59i)47-s + ⋯
L(s)  = 1  + (0.00646 − 0.0112i)5-s + (−0.645 − 0.763i)7-s + (0.567 + 0.982i)11-s − 0.824·13-s + (−0.171 − 0.297i)17-s + (−0.874 + 1.51i)19-s + (0.622 − 1.07i)23-s + (0.499 + 0.865i)25-s − 1.85·29-s + (0.768 + 1.33i)31-s + (−0.0127 + 0.00229i)35-s + (−0.121 + 0.209i)37-s + 0.685·41-s + 0.573·43-s + (−0.302 + 0.524i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.239 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.239 - 0.970i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (1297, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1512,\ (\ :1/2),\ -0.239 - 0.970i)\)
\(L(1)\)  \(\approx\)  \(0.8895306996\)
\(L(\frac12)\)  \(\approx\)  \(0.8895306996\)
\(L(\frac{3}{2})\)   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 \( 1 + (1.70 + 2.02i)T \)
good5 \( 1 + (-0.0144 + 0.0250i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.88 - 3.25i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.97T + 13T^{2} \)
17 \( 1 + (0.708 + 1.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.81 - 6.60i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.98 + 5.17i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.99T + 29T^{2} \)
31 \( 1 + (-4.27 - 7.40i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.737 - 1.27i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.38T + 41T^{2} \)
43 \( 1 - 3.76T + 43T^{2} \)
47 \( 1 + (2.07 - 3.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.60 + 6.24i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.93 - 12.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.61 - 9.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.77 + 4.81i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.5T + 71T^{2} \)
73 \( 1 + (-2.17 - 3.77i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.04 - 12.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.62T + 83T^{2} \)
89 \( 1 + (2.14 - 3.71i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.77T + 97T^{2} \)
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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

−9.705385670901680435923015798383, −9.081383560294110610946821644539, −8.022806864723203879376457258104, −7.09471171389538140680295738181, −6.73976428513026946388947007155, −5.60263383911123579316027586646, −4.53155224880954974589001790614, −3.86128369527737805081271873833, −2.68840482779247008945762116530, −1.40892832868903719392089337375, 0.35009753323871427038345690404, 2.16755956619563328627590355335, 3.05022498803608971695829725126, 4.09939175809969117759853657769, 5.17302168499528893001031189931, 6.04202207029127892041718602442, 6.68567455488451140995857236702, 7.63285613292140802056349081074, 8.638546521747038697644253899761, 9.241041341439582684078463749901

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