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Properties

Label 2-525-105.32-c1-0-14
Degree $2$
Conductor $525$
Sign $0.390 - 0.920i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.17 − 0.582i)2-s + (0.245 + 1.71i)3-s + (2.64 + 1.52i)4-s + (0.465 − 3.86i)6-s + (2.38 + 1.15i)7-s + (−1.68 − 1.68i)8-s + (−2.87 + 0.840i)9-s + (3.88 + 2.24i)11-s + (−1.97 + 4.91i)12-s + (1.08 − 1.08i)13-s + (−4.50 − 3.88i)14-s + (−0.381 − 0.660i)16-s + (0.548 + 2.04i)17-s + (6.74 − 0.150i)18-s + (3.66 − 2.11i)19-s + ⋯
L(s)  = 1  + (−1.53 − 0.411i)2-s + (0.141 + 0.989i)3-s + (1.32 + 0.764i)4-s + (0.189 − 1.57i)6-s + (0.900 + 0.435i)7-s + (−0.595 − 0.595i)8-s + (−0.959 + 0.280i)9-s + (1.17 + 0.676i)11-s + (−0.569 + 1.41i)12-s + (0.300 − 0.300i)13-s + (−1.20 − 1.03i)14-s + (−0.0952 − 0.165i)16-s + (0.133 + 0.496i)17-s + (1.58 − 0.0354i)18-s + (0.839 − 0.484i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.390 - 0.920i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.390 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.677061 + 0.448163i\)
\(L(\frac12)\) \(\approx\) \(0.677061 + 0.448163i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.245 - 1.71i)T \)
5 \( 1 \)
7 \( 1 + (-2.38 - 1.15i)T \)
good2 \( 1 + (2.17 + 0.582i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-3.88 - 2.24i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.08 + 1.08i)T - 13iT^{2} \)
17 \( 1 + (-0.548 - 2.04i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.66 + 2.11i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.840 + 3.13i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 1.69T + 29T^{2} \)
31 \( 1 + (-0.530 + 0.918i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.54 - 5.75i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 5.84iT - 41T^{2} \)
43 \( 1 + (2.00 - 2.00i)T - 43iT^{2} \)
47 \( 1 + (5.10 + 1.36i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-8.34 + 2.23i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.35 - 4.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.88 - 6.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.569 - 0.152i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.66iT - 71T^{2} \)
73 \( 1 + (1.13 + 4.22i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.78 - 3.33i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.0 - 11.0i)T + 83iT^{2} \)
89 \( 1 + (1.75 + 3.04i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.60 - 5.60i)T + 97iT^{2} \)
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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

−10.81908474998618558871850784271, −9.967339712583146705773064770505, −9.329282899215422275268531167040, −8.583507813511189082131888973311, −7.970902535756472350488801806722, −6.75167015259554952977554550064, −5.31768415526723817757367908985, −4.24043954772444364177738176855, −2.79520957365616691243255007416, −1.43835276565815336442789332011, 0.914857810153147102198000749366, 1.80657532590581210020067655896, 3.68045370663721007432640211465, 5.52281595988044870987348994994, 6.59747207558499906783070801026, 7.32915924811306152709847915303, 7.993543044729148025998011398889, 8.831608341994321736250542383551, 9.411031717449556766756657232987, 10.66214986145799640212135452273

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