Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $-0.652 + 0.757i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.99 − 3.45i)5-s + (−0.657 + 2.56i)7-s + (−0.444 − 0.769i)11-s − 6.98·13-s + (−0.342 − 0.592i)17-s + (1.73 − 3.00i)19-s + (2.94 − 5.10i)23-s + (−5.43 − 9.42i)25-s − 5.87·29-s + (−0.604 − 1.04i)31-s + (7.53 + 7.37i)35-s + (4.32 − 7.49i)37-s − 6.30·41-s + 0.888·43-s + (1.59 − 2.76i)47-s + ⋯
L(s)  = 1  + (0.890 − 1.54i)5-s + (−0.248 + 0.968i)7-s + (−0.133 − 0.231i)11-s − 1.93·13-s + (−0.0830 − 0.143i)17-s + (0.397 − 0.688i)19-s + (0.614 − 1.06i)23-s + (−1.08 − 1.88i)25-s − 1.09·29-s + (−0.108 − 0.187i)31-s + (1.27 + 1.24i)35-s + (0.711 − 1.23i)37-s − 0.983·41-s + 0.135·43-s + (0.232 − 0.402i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.652 + 0.757i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (1297, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1512,\ (\ :1/2),\ -0.652 + 0.757i)\)
\(L(1)\)  \(\approx\)  \(1.177403564\)
\(L(\frac12)\)  \(\approx\)  \(1.177403564\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.657 - 2.56i)T \)
good5 \( 1 + (-1.99 + 3.45i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.444 + 0.769i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.98T + 13T^{2} \)
17 \( 1 + (0.342 + 0.592i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.73 + 3.00i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.94 + 5.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.87T + 29T^{2} \)
31 \( 1 + (0.604 + 1.04i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.32 + 7.49i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.30T + 41T^{2} \)
43 \( 1 - 0.888T + 43T^{2} \)
47 \( 1 + (-1.59 + 2.76i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.890 + 1.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.11 - 8.86i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.882 + 1.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.83 + 10.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.50T + 71T^{2} \)
73 \( 1 + (-5.14 - 8.90i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.64 - 4.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + (-5.77 + 9.99i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.6T + 97T^{2} \)
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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

−9.298398128183196579422865936012, −8.645178241393765526401188851820, −7.68487603642934090988859290628, −6.67468001195173103738090832250, −5.53672476570280026949315691089, −5.21981916160376252834429497907, −4.40393227275553267756799940269, −2.72186093701780133438480731575, −2.00973458047065826403460889663, −0.42607580657531698654633377937, 1.74960265600451240028690125583, 2.79901954889025610213546905222, 3.56064257024323296044012618605, 4.84063605641440421295822524988, 5.75821417713250053771114799989, 6.71602198177200628826614317213, 7.28089779365576087204650702801, 7.76076196420807806959476240186, 9.403051863299899659162846374044, 9.878168185629579528257908527180

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