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Properties

Label 2-525-5.3-c2-0-6
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  + (−2.08 + 2.08i)2-s + (1.22 + 1.22i)3-s − 4.73i·4-s − 5.11·6-s + (1.87 − 1.87i)7-s + (1.53 + 1.53i)8-s + 2.99i·9-s + 2.70·11-s + (5.79 − 5.79i)12-s + (2.37 + 2.37i)13-s + 7.81i·14-s + 12.5·16-s + (−16.3 + 16.3i)17-s + (−6.26 − 6.26i)18-s + 9.18i·19-s + ⋯
L(s)  = 1  + (−1.04 + 1.04i)2-s + (0.408 + 0.408i)3-s − 1.18i·4-s − 0.853·6-s + (0.267 − 0.267i)7-s + (0.191 + 0.191i)8-s + 0.333i·9-s + 0.245·11-s + (0.483 − 0.483i)12-s + (0.183 + 0.183i)13-s + 0.558i·14-s + 0.782·16-s + (−0.963 + 0.963i)17-s + (−0.348 − 0.348i)18-s + 0.483i·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

\(L(\frac{3}{2})\) \(\approx\) \(0.5807083270\)
\(L(\frac12)\) \(\approx\) \(0.5807083270\)
\(L(2)\) 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 \( 1 + (2.08 - 2.08i)T - 4iT^{2} \)
11 \( 1 - 2.70T + 121T^{2} \)
13 \( 1 + (-2.37 - 2.37i)T + 169iT^{2} \)
17 \( 1 + (16.3 - 16.3i)T - 289iT^{2} \)
19 \( 1 - 9.18iT - 361T^{2} \)
23 \( 1 + (21.4 + 21.4i)T + 529iT^{2} \)
29 \( 1 - 52.3iT - 841T^{2} \)
31 \( 1 + 5.01T + 961T^{2} \)
37 \( 1 + (23.2 - 23.2i)T - 1.36e3iT^{2} \)
41 \( 1 + 60.5T + 1.68e3T^{2} \)
43 \( 1 + (-8.78 - 8.78i)T + 1.84e3iT^{2} \)
47 \( 1 + (2.24 - 2.24i)T - 2.20e3iT^{2} \)
53 \( 1 + (-25.6 - 25.6i)T + 2.80e3iT^{2} \)
59 \( 1 - 100. iT - 3.48e3T^{2} \)
61 \( 1 + 82.1T + 3.72e3T^{2} \)
67 \( 1 + (-65.1 + 65.1i)T - 4.48e3iT^{2} \)
71 \( 1 + 22.8T + 5.04e3T^{2} \)
73 \( 1 + (-5.38 - 5.38i)T + 5.32e3iT^{2} \)
79 \( 1 + 117. iT - 6.24e3T^{2} \)
83 \( 1 + (85.5 + 85.5i)T + 6.88e3iT^{2} \)
89 \( 1 + 119. iT - 7.92e3T^{2} \)
97 \( 1 + (-55.1 + 55.1i)T - 9.40e3iT^{2} \)
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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

−10.55837229397186408598267506411, −10.25251681356456081617162944645, −8.938497183396385450229291139946, −8.662504809009083170975083051616, −7.73562421422359059750175937567, −6.79922462110059914261558411901, −5.99396379055288407548843430435, −4.65279995487332342369490170860, −3.49017606361996875884056799502, −1.63756380644655128050991220642, 0.30267741339306234534718757321, 1.77545076985454436374580989139, 2.63762948095306400211557698622, 3.88873869865763307629815746962, 5.43550990085317475789781187017, 6.73402109488038080816517364929, 7.83205947869887975411406925063, 8.532923667512336099640829530629, 9.358674473534458333203453490087, 9.945387543412066223050056672088

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