Properties

Degree $2$
Conductor $525$
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  − 3i·2-s + 3i·3-s − 4-s + 9·6-s + 7i·7-s − 21i·8-s − 9·9-s − 36·11-s − 3i·12-s + 34i·13-s + 21·14-s − 71·16-s + 42i·17-s + 27i·18-s + 124·19-s + ⋯
L(s)  = 1  − 1.06i·2-s + 0.577i·3-s − 0.125·4-s + 0.612·6-s + 0.377i·7-s − 0.928i·8-s − 0.333·9-s − 0.986·11-s − 0.0721i·12-s + 0.725i·13-s + 0.400·14-s − 1.10·16-s + 0.599i·17-s + 0.353i·18-s + 1.49·19-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

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.447 - 0.894i$
Motivic weight: \(3\)
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :3/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.199350181\)
\(L(\frac12)\) \(\approx\) \(1.199350181\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3iT \)
5 \( 1 \)
7 \( 1 - 7iT \)
good2 \( 1 + 3iT - 8T^{2} \)
11 \( 1 + 36T + 1.33e3T^{2} \)
13 \( 1 - 34iT - 2.19e3T^{2} \)
17 \( 1 - 42iT - 4.91e3T^{2} \)
19 \( 1 - 124T + 6.85e3T^{2} \)
23 \( 1 - 1.21e4T^{2} \)
29 \( 1 + 102T + 2.43e4T^{2} \)
31 \( 1 + 160T + 2.97e4T^{2} \)
37 \( 1 - 398iT - 5.06e4T^{2} \)
41 \( 1 + 318T + 6.89e4T^{2} \)
43 \( 1 - 268iT - 7.95e4T^{2} \)
47 \( 1 - 240iT - 1.03e5T^{2} \)
53 \( 1 - 498iT - 1.48e5T^{2} \)
59 \( 1 - 132T + 2.05e5T^{2} \)
61 \( 1 - 398T + 2.26e5T^{2} \)
67 \( 1 - 92iT - 3.00e5T^{2} \)
71 \( 1 + 720T + 3.57e5T^{2} \)
73 \( 1 - 502iT - 3.89e5T^{2} \)
79 \( 1 - 1.02e3T + 4.93e5T^{2} \)
83 \( 1 - 204iT - 5.71e5T^{2} \)
89 \( 1 + 354T + 7.04e5T^{2} \)
97 \( 1 + 286iT - 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.66193279970819223325488061387, −9.865879719944274928991822775243, −9.251331029544100009051416102202, −8.090003120939171846755625404253, −6.99400254499154043103699204717, −5.78029321645497964324701007115, −4.73330100481057940440338863548, −3.55039658846958212328398534473, −2.69734339914058008430854870133, −1.45223355996482604762642215167, 0.34937687866765288249035281407, 2.11091259680074434324615154753, 3.37123498533767392642043421118, 5.23618932718648716172114894495, 5.56199206591429851803549688016, 6.91869649410433302313298807450, 7.45558017372683787280519828667, 8.079302956043925042421468521326, 9.143439904311811354957570134750, 10.32257245525055849242410444051

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