Properties

Label 2-525-5.3-c2-0-33
Degree $2$
Conductor $525$
Sign $-0.850 + 0.525i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.72 − 2.72i)2-s + (1.22 + 1.22i)3-s − 10.8i·4-s + 6.67·6-s + (1.87 − 1.87i)7-s + (−18.6 − 18.6i)8-s + 2.99i·9-s + 3.42·11-s + (13.2 − 13.2i)12-s + (−7.98 − 7.98i)13-s − 10.1i·14-s − 57.9·16-s + (16.5 − 16.5i)17-s + (8.16 + 8.16i)18-s − 1.38i·19-s + ⋯
L(s)  = 1  + (1.36 − 1.36i)2-s + (0.408 + 0.408i)3-s − 2.70i·4-s + 1.11·6-s + (0.267 − 0.267i)7-s + (−2.32 − 2.32i)8-s + 0.333i·9-s + 0.311·11-s + (1.10 − 1.10i)12-s + (−0.613 − 0.613i)13-s − 0.727i·14-s − 3.62·16-s + (0.974 − 0.974i)17-s + (0.453 + 0.453i)18-s − 0.0726i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ -0.850 + 0.525i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.005884806\)
\(L(\frac12)\) \(\approx\) \(4.005884806\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 \)
7 \( 1 + (-1.87 + 1.87i)T \)
good2 \( 1 + (-2.72 + 2.72i)T - 4iT^{2} \)
11 \( 1 - 3.42T + 121T^{2} \)
13 \( 1 + (7.98 + 7.98i)T + 169iT^{2} \)
17 \( 1 + (-16.5 + 16.5i)T - 289iT^{2} \)
19 \( 1 + 1.38iT - 361T^{2} \)
23 \( 1 + (18.8 + 18.8i)T + 529iT^{2} \)
29 \( 1 - 45.7iT - 841T^{2} \)
31 \( 1 - 43.2T + 961T^{2} \)
37 \( 1 + (2.18 - 2.18i)T - 1.36e3iT^{2} \)
41 \( 1 - 6.61T + 1.68e3T^{2} \)
43 \( 1 + (-44.1 - 44.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (14.2 - 14.2i)T - 2.20e3iT^{2} \)
53 \( 1 + (-44.4 - 44.4i)T + 2.80e3iT^{2} \)
59 \( 1 + 17.2iT - 3.48e3T^{2} \)
61 \( 1 - 48.0T + 3.72e3T^{2} \)
67 \( 1 + (40.9 - 40.9i)T - 4.48e3iT^{2} \)
71 \( 1 + 38.7T + 5.04e3T^{2} \)
73 \( 1 + (66.4 + 66.4i)T + 5.32e3iT^{2} \)
79 \( 1 - 5.30iT - 6.24e3T^{2} \)
83 \( 1 + (-62.5 - 62.5i)T + 6.88e3iT^{2} \)
89 \( 1 + 44.8iT - 7.92e3T^{2} \)
97 \( 1 + (-30.5 + 30.5i)T - 9.40e3iT^{2} \)
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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

−10.32705607055432672489371059408, −9.957099019038986823946143466414, −8.900403765242470141378941523911, −7.47971273760915115199567173355, −6.13553274253002003647711800131, −5.07682330027665699822834873933, −4.42769806556475298581281125389, −3.30330083943862557618498802173, −2.51931713281667663385704557420, −1.01760860847430405089574917743, 2.27125561871507374063084339132, 3.61075389143783224358994609162, 4.42906900519352045193853128810, 5.64166101764699724266167847391, 6.28199543308253541121587811479, 7.31980099901661029812077915900, 7.961320250296825319571962422250, 8.722972991910745258004535785438, 9.930976580993133054585323014537, 11.78861957711447599953882759636

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