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Properties

Label 2-525-5.4-c5-0-18
Degree $2$
Conductor $525$
Sign $-0.894 - 0.447i$
Analytic cond. $84.2015$
Root an. cond. $9.17613$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.76i·2-s − 9i·3-s + 17.8·4-s + 33.8·6-s + 49i·7-s + 187. i·8-s − 81·9-s + 277.·11-s − 160. i·12-s + 569. i·13-s − 184.·14-s − 133.·16-s + 181. i·17-s − 304. i·18-s − 970.·19-s + ⋯
L(s)  = 1  + 0.664i·2-s − 0.577i·3-s + 0.558·4-s + 0.383·6-s + 0.377i·7-s + 1.03i·8-s − 0.333·9-s + 0.690·11-s − 0.322i·12-s + 0.934i·13-s − 0.251·14-s − 0.130·16-s + 0.152i·17-s − 0.221i·18-s − 0.616·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(84.2015\)
Root analytic conductor: \(9.17613\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :5/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.666071428\)
\(L(\frac12)\) \(\approx\) \(1.666071428\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9iT \)
5 \( 1 \)
7 \( 1 - 49iT \)
good2 \( 1 - 3.76iT - 32T^{2} \)
11 \( 1 - 277.T + 1.61e5T^{2} \)
13 \( 1 - 569. iT - 3.71e5T^{2} \)
17 \( 1 - 181. iT - 1.41e6T^{2} \)
19 \( 1 + 970.T + 2.47e6T^{2} \)
23 \( 1 - 2.00e3iT - 6.43e6T^{2} \)
29 \( 1 + 328.T + 2.05e7T^{2} \)
31 \( 1 + 2.63e3T + 2.86e7T^{2} \)
37 \( 1 + 7.10e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.47e4T + 1.15e8T^{2} \)
43 \( 1 + 2.55e3iT - 1.47e8T^{2} \)
47 \( 1 + 5.33e3iT - 2.29e8T^{2} \)
53 \( 1 - 87.4iT - 4.18e8T^{2} \)
59 \( 1 - 4.80e3T + 7.14e8T^{2} \)
61 \( 1 + 1.97e4T + 8.44e8T^{2} \)
67 \( 1 + 1.82e3iT - 1.35e9T^{2} \)
71 \( 1 + 3.26e4T + 1.80e9T^{2} \)
73 \( 1 - 7.78e4iT - 2.07e9T^{2} \)
79 \( 1 - 3.83e4T + 3.07e9T^{2} \)
83 \( 1 + 2.07e4iT - 3.93e9T^{2} \)
89 \( 1 - 4.87e4T + 5.58e9T^{2} \)
97 \( 1 - 8.63e4iT - 8.58e9T^{2} \)
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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

−10.56875690767931605289700763068, −9.257736809024924893618707385061, −8.544667802780720944979189585127, −7.55035014808827730383043805007, −6.78229027767034776403758544546, −6.13562113008895180223169089072, −5.16263427190186160369863369813, −3.72916651025742857953116442402, −2.31208028015445244166891406283, −1.49555622783450661680113572562, 0.33944150571486145953012140117, 1.57875726260710994970242581411, 2.85648949273743909416498225058, 3.69477942760246546393659325973, 4.72946055210468264636222661566, 6.04595841068478862869498349280, 6.87367989773652978853613018346, 7.968377752394385028806679420174, 9.017525654375270698899748735283, 10.02735136699110240720892364309

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