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Properties

Label 2-525-5.4-c3-0-31
Degree $2$
Conductor $525$
Sign $0.894 - 0.447i$
Analytic cond. $30.9760$
Root an. cond. $5.56560$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.60i·2-s + 3i·3-s − 13.1·4-s − 13.8·6-s − 7i·7-s − 23.8i·8-s − 9·9-s − 52.9·11-s − 39.5i·12-s + 19.6i·13-s + 32.2·14-s + 4.19·16-s − 61.5i·17-s − 41.4i·18-s − 27.0·19-s + ⋯
L(s)  = 1  + 1.62i·2-s + 0.577i·3-s − 1.64·4-s − 0.939·6-s − 0.377i·7-s − 1.05i·8-s − 0.333·9-s − 1.45·11-s − 0.950i·12-s + 0.418i·13-s + 0.614·14-s + 0.0655·16-s − 0.877i·17-s − 0.542i·18-s − 0.327·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(30.9760\)
Root analytic conductor: \(5.56560\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :3/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6687997718\)
\(L(\frac12)\) \(\approx\) \(0.6687997718\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3iT \)
5 \( 1 \)
7 \( 1 + 7iT \)
good2 \( 1 - 4.60iT - 8T^{2} \)
11 \( 1 + 52.9T + 1.33e3T^{2} \)
13 \( 1 - 19.6iT - 2.19e3T^{2} \)
17 \( 1 + 61.5iT - 4.91e3T^{2} \)
19 \( 1 + 27.0T + 6.85e3T^{2} \)
23 \( 1 - 19.2iT - 1.21e4T^{2} \)
29 \( 1 - 167.T + 2.43e4T^{2} \)
31 \( 1 - 225.T + 2.97e4T^{2} \)
37 \( 1 + 311. iT - 5.06e4T^{2} \)
41 \( 1 - 12.8T + 6.89e4T^{2} \)
43 \( 1 + 114. iT - 7.95e4T^{2} \)
47 \( 1 - 207. iT - 1.03e5T^{2} \)
53 \( 1 - 227. iT - 1.48e5T^{2} \)
59 \( 1 - 605.T + 2.05e5T^{2} \)
61 \( 1 + 315.T + 2.26e5T^{2} \)
67 \( 1 + 720. iT - 3.00e5T^{2} \)
71 \( 1 + 56.2T + 3.57e5T^{2} \)
73 \( 1 + 1.15e3iT - 3.89e5T^{2} \)
79 \( 1 + 1.15e3T + 4.93e5T^{2} \)
83 \( 1 - 692. iT - 5.71e5T^{2} \)
89 \( 1 + 1.41e3T + 7.04e5T^{2} \)
97 \( 1 + 661. iT - 9.12e5T^{2} \)
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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

−10.29402925851733593654544266171, −9.408755948186620149302985636123, −8.480915701443864452877461889894, −7.73525096028459474663624955667, −6.92731980661562678306430155682, −5.90364123828992053120050038772, −5.01857864884309908863886795825, −4.31457445569062469123766278194, −2.71419343902695543145047513588, −0.22928252514600446196116189896, 1.10329893720963134662764056716, 2.38545291689102615255600580106, 3.02198993942106045084041308269, 4.42448211695298155051868228403, 5.48686849013546788458781785594, 6.70281487428369699542783606952, 8.150957866751937998577628497728, 8.569396878179606032346868009331, 10.06195197210861094800100264199, 10.30752121487182037643996835336

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