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Properties

Label 2-525-5.2-c2-0-14
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: $\chi_{525} (232, \cdot )$
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

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

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