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Properties

Label 2-525-5.4-c1-0-2
Degree $2$
Conductor $525$
Sign $-0.447 + 0.894i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.23i·2-s + i·3-s − 3.00·4-s − 2.23·6-s + i·7-s − 2.23i·8-s − 9-s − 2.47·11-s − 3.00i·12-s + 4.47i·13-s − 2.23·14-s − 0.999·16-s − 2i·17-s − 2.23i·18-s − 6.47·19-s + ⋯
L(s)  = 1  + 1.58i·2-s + 0.577i·3-s − 1.50·4-s − 0.912·6-s + 0.377i·7-s − 0.790i·8-s − 0.333·9-s − 0.745·11-s − 0.866i·12-s + 1.24i·13-s − 0.597·14-s − 0.249·16-s − 0.485i·17-s − 0.527i·18-s − 1.48·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.470093 - 0.760627i\)
\(L(\frac12)\) \(\approx\) \(0.470093 - 0.760627i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 \)
7 \( 1 - iT \)
good2 \( 1 - 2.23iT - 2T^{2} \)
11 \( 1 + 2.47T + 11T^{2} \)
13 \( 1 - 4.47iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 6.47T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 - 10.9iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 8.94iT - 43T^{2} \)
47 \( 1 + 4.94iT - 47T^{2} \)
53 \( 1 - 12.4iT - 53T^{2} \)
59 \( 1 + 8.94T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 - 3.52iT - 73T^{2} \)
79 \( 1 - 4.94T + 79T^{2} \)
83 \( 1 + 0.944iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + 0.472iT - 97T^{2} \)
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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

−11.38895922061054819077392864895, −10.34584014667418870707582443000, −9.384429634263968409259286102848, −8.543114252437377095255904490900, −7.981267605299237259145031309138, −6.63382353458533584844960279022, −6.23727846449937061979343872709, −4.87466340144551585498955815265, −4.45622046265954078035905775448, −2.59911001528112120346292553136, 0.50894266224716562366475865466, 1.99797892323627015853310780207, 3.00945220165908708346934988104, 4.10166618484140994166235037567, 5.33844204460762487397124468854, 6.57584905763154877516153030602, 7.84016183007058400081124506286, 8.554480792995502720289426196005, 9.758285962734796988999211494058, 10.58074097190601166831850132836

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