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  + 1.82i·2-s + 3i·3-s + 4.65·4-s − 5.48·6-s − 7i·7-s + 23.1i·8-s − 9·9-s − 64.5·11-s + 13.9i·12-s + 32.3i·13-s + 12.7·14-s − 5.05·16-s − 56.3i·17-s − 16.4i·18-s + 2.74·19-s + ⋯
L(s)  = 1  + 0.646i·2-s + 0.577i·3-s + 0.582·4-s − 0.373·6-s − 0.377i·7-s + 1.02i·8-s − 0.333·9-s − 1.76·11-s + 0.336i·12-s + 0.690i·13-s + 0.244·14-s − 0.0790·16-s − 0.803i·17-s − 0.215i·18-s + 0.0331·19-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  =  \((2,\ 525,\ (\ :3/2),\ -0.447 + 0.894i)\)
\(L(2)\)  \(\approx\)  \(0.2904210379\)
\(L(\frac12)\)  \(\approx\)  \(0.2904210379\)
\(L(\frac{5}{2})\)   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 \( 1 - 1.82iT - 8T^{2} \)
11 \( 1 + 64.5T + 1.33e3T^{2} \)
13 \( 1 - 32.3iT - 2.19e3T^{2} \)
17 \( 1 + 56.3iT - 4.91e3T^{2} \)
19 \( 1 - 2.74T + 6.85e3T^{2} \)
23 \( 1 + 88.1iT - 1.21e4T^{2} \)
29 \( 1 + 246.T + 2.43e4T^{2} \)
31 \( 1 + 110.T + 2.97e4T^{2} \)
37 \( 1 - 120. iT - 5.06e4T^{2} \)
41 \( 1 + 176.T + 6.89e4T^{2} \)
43 \( 1 - 443. iT - 7.95e4T^{2} \)
47 \( 1 + 345. iT - 1.03e5T^{2} \)
53 \( 1 + 260. iT - 1.48e5T^{2} \)
59 \( 1 + 628.T + 2.05e5T^{2} \)
61 \( 1 + 115.T + 2.26e5T^{2} \)
67 \( 1 + 951. iT - 3.00e5T^{2} \)
71 \( 1 - 356.T + 3.57e5T^{2} \)
73 \( 1 - 656. iT - 3.89e5T^{2} \)
79 \( 1 + 440.T + 4.93e5T^{2} \)
83 \( 1 - 54.4iT - 5.71e5T^{2} \)
89 \( 1 - 1.01e3T + 7.04e5T^{2} \)
97 \( 1 + 724. iT - 9.12e5T^{2} \)
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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

−10.96571478438731095558975337879, −10.26480403585386285440952806773, −9.261538804881488280173532383081, −8.130523941194192971520879909234, −7.48416705526534003873852523555, −6.53901162618826522119014304450, −5.43645285244259870090673784417, −4.73575836325827677520876230848, −3.21729990458931298399777354219, −2.12008348659027572228565817607, 0.07683288953441671884117443745, 1.68404896701510478572898027533, 2.60844670117902808385764646680, 3.58594363017692517235905469043, 5.34928425951295273483486321233, 6.01219921610400689026852646202, 7.38587612940060796665089327720, 7.79597686974351444912605466569, 9.016646242383752223955638437341, 10.19447266995388962490466429262

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