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Properties

Label 2-525-175.153-c1-0-26
Degree $2$
Conductor $525$
Sign $0.103 + 0.994i$
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  + (−0.236 − 0.463i)2-s + (−0.987 + 0.156i)3-s + (1.01 − 1.39i)4-s + (1.07 − 1.95i)5-s + (0.305 + 0.420i)6-s + (2.24 + 1.39i)7-s + (−1.91 − 0.303i)8-s + (0.951 − 0.309i)9-s + (−1.16 − 0.0364i)10-s + (0.894 − 2.75i)11-s + (−0.785 + 1.54i)12-s + (5.31 + 2.70i)13-s + (0.116 − 1.37i)14-s + (−0.757 + 2.10i)15-s + (−0.757 − 2.33i)16-s + (−0.573 − 0.0908i)17-s + ⋯
L(s)  = 1  + (−0.166 − 0.327i)2-s + (−0.570 + 0.0903i)3-s + (0.508 − 0.699i)4-s + (0.481 − 0.876i)5-s + (0.124 + 0.171i)6-s + (0.849 + 0.527i)7-s + (−0.677 − 0.107i)8-s + (0.317 − 0.103i)9-s + (−0.367 − 0.0115i)10-s + (0.269 − 0.830i)11-s + (−0.226 + 0.444i)12-s + (1.47 + 0.751i)13-s + (0.0311 − 0.366i)14-s + (−0.195 + 0.543i)15-s + (−0.189 − 0.582i)16-s + (−0.139 − 0.0220i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.103 + 0.994i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (328, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.103 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09736 - 0.988978i\)
\(L(\frac12)\) \(\approx\) \(1.09736 - 0.988978i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.987 - 0.156i)T \)
5 \( 1 + (-1.07 + 1.95i)T \)
7 \( 1 + (-2.24 - 1.39i)T \)
good2 \( 1 + (0.236 + 0.463i)T + (-1.17 + 1.61i)T^{2} \)
11 \( 1 + (-0.894 + 2.75i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-5.31 - 2.70i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (0.573 + 0.0908i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (5.15 - 3.74i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-4.49 + 2.28i)T + (13.5 - 18.6i)T^{2} \)
29 \( 1 + (0.540 - 0.743i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.38 + 4.65i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-4.09 - 2.08i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (1.35 - 0.441i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (2.90 + 2.90i)T + 43iT^{2} \)
47 \( 1 + (1.44 + 9.12i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (0.0105 + 0.0667i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (1.28 + 3.95i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.48 - 1.45i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (0.433 - 2.73i)T + (-63.7 - 20.7i)T^{2} \)
71 \( 1 + (-0.337 - 0.245i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.52 - 4.96i)T + (-42.9 + 59.0i)T^{2} \)
79 \( 1 + (3.62 - 4.99i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.72 - 10.8i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (5.17 - 15.9i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-0.749 - 4.73i)T + (-92.2 + 29.9i)T^{2} \)
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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

−10.95316887168465912746435600733, −9.852426887504576567931488906773, −8.841537639797587145665215712953, −8.397953946862123071825592553296, −6.61199750872809535756903722142, −5.94680124515711855392922772885, −5.22746768513162385827808094242, −4.00202786676586958337012128266, −2.06488708511854170933635876707, −1.11058864720381664383888451188, 1.73309188922863507059666312003, 3.16877088068067596192050167926, 4.40964161541180099485148971230, 5.78018220349564902227919245206, 6.66979611331874412178360463163, 7.26310917395000150131634702060, 8.192078456947510952462523936362, 9.206652906029210921608303894532, 10.61360905831746545065241278137, 10.95052021418624306908531135216

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