Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $-0.447 + 0.894i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.82i·2-s + 3i·3-s − 6.65·4-s + 11.4·6-s − 7i·7-s − 5.14i·8-s − 9·9-s + 48.5·11-s − 19.9i·12-s + 43.6i·13-s − 26.7·14-s − 72.9·16-s − 67.6i·17-s + 34.4i·18-s + 93.2·19-s + ⋯
L(s)  = 1  − 1.35i·2-s + 0.577i·3-s − 0.832·4-s + 0.781·6-s − 0.377i·7-s − 0.227i·8-s − 0.333·9-s + 1.33·11-s − 0.480i·12-s + 0.931i·13-s − 0.511·14-s − 1.13·16-s − 0.965i·17-s + 0.451i·18-s + 1.12·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.447 + 0.894i$
motivic weight  =  \(3\)
character  :  $\chi_{525} (274, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :3/2),\ -0.447 + 0.894i)\)
\(L(2)\)  \(\approx\)  \(2.065439991\)
\(L(\frac12)\)  \(\approx\)  \(2.065439991\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - 3iT \)
5 \( 1 \)
7 \( 1 + 7iT \)
good2 \( 1 + 3.82iT - 8T^{2} \)
11 \( 1 - 48.5T + 1.33e3T^{2} \)
13 \( 1 - 43.6iT - 2.19e3T^{2} \)
17 \( 1 + 67.6iT - 4.91e3T^{2} \)
19 \( 1 - 93.2T + 6.85e3T^{2} \)
23 \( 1 - 104. iT - 1.21e4T^{2} \)
29 \( 1 - 58.7T + 2.43e4T^{2} \)
31 \( 1 + 9.08T + 2.97e4T^{2} \)
37 \( 1 + 252. iT - 5.06e4T^{2} \)
41 \( 1 - 276.T + 6.89e4T^{2} \)
43 \( 1 - 92.6iT - 7.95e4T^{2} \)
47 \( 1 + 582. iT - 1.03e5T^{2} \)
53 \( 1 + 623. iT - 1.48e5T^{2} \)
59 \( 1 - 524.T + 2.05e5T^{2} \)
61 \( 1 + 352.T + 2.26e5T^{2} \)
67 \( 1 + 736. iT - 3.00e5T^{2} \)
71 \( 1 + 492.T + 3.57e5T^{2} \)
73 \( 1 + 1.16e3iT - 3.89e5T^{2} \)
79 \( 1 - 872.T + 4.93e5T^{2} \)
83 \( 1 - 529. iT - 5.71e5T^{2} \)
89 \( 1 - 385.T + 7.04e5T^{2} \)
97 \( 1 + 463. iT - 9.12e5T^{2} \)
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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

−10.18508906619197874500035017403, −9.409642625097033563853623230644, −9.067591099767092860728007619039, −7.42352238183458281935175914184, −6.52098345512566227270857767548, −5.04721798697159176931049261675, −3.99460517101096099218866406134, −3.35557173768033699089864519082, −1.95636227347155551477786825347, −0.74753662772118604501513033014, 1.17559999782369849778827296974, 2.80115695381813566034433924974, 4.34444997278207465325877110384, 5.61380325139987437745669345677, 6.21190783877143353400978157624, 7.03224188546199446463504659677, 7.935307784766652750295654049140, 8.601004518460505581853783004966, 9.478966924087255398068449051660, 10.76250793445543526028858476375

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