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Properties

Label 2-525-25.16-c1-0-0
Degree $2$
Conductor $525$
Sign $0.283 - 0.958i$
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  + (−2.01 − 1.46i)2-s + (0.309 + 0.951i)3-s + (1.29 + 3.98i)4-s + (−1.44 − 1.70i)5-s + (0.768 − 2.36i)6-s − 7-s + (1.68 − 5.17i)8-s + (−0.809 + 0.587i)9-s + (0.422 + 5.54i)10-s + (−0.214 − 0.155i)11-s + (−3.38 + 2.46i)12-s + (1.25 − 0.914i)13-s + (2.01 + 1.46i)14-s + (1.17 − 1.90i)15-s + (−4.18 + 3.03i)16-s + (0.612 − 1.88i)17-s + ⋯
L(s)  = 1  + (−1.42 − 1.03i)2-s + (0.178 + 0.549i)3-s + (0.647 + 1.99i)4-s + (−0.647 − 0.762i)5-s + (0.313 − 0.965i)6-s − 0.377·7-s + (0.594 − 1.83i)8-s + (−0.269 + 0.195i)9-s + (0.133 + 1.75i)10-s + (−0.0647 − 0.0470i)11-s + (−0.978 + 0.710i)12-s + (0.349 − 0.253i)13-s + (0.537 + 0.390i)14-s + (0.302 − 0.491i)15-s + (−1.04 + 0.759i)16-s + (0.148 − 0.456i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.283 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.283 - 0.958i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.283 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.220590 + 0.164735i\)
\(L(\frac12)\) \(\approx\) \(0.220590 + 0.164735i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (1.44 + 1.70i)T \)
7 \( 1 + T \)
good2 \( 1 + (2.01 + 1.46i)T + (0.618 + 1.90i)T^{2} \)
11 \( 1 + (0.214 + 0.155i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.25 + 0.914i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.612 + 1.88i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.44 - 4.44i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (3.31 + 2.41i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.89 - 8.92i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.569 - 1.75i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (8.66 - 6.29i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (8.64 - 6.27i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 0.340T + 43T^{2} \)
47 \( 1 + (-0.0625 - 0.192i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.55 - 7.86i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.85 + 1.35i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-7.35 - 5.34i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-2.88 + 8.88i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-3.11 - 9.59i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.04 - 0.762i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.18 + 6.72i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.57 - 4.85i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (2.89 + 2.10i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (1.37 + 4.23i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.70367152121727443751091672747, −10.19916661422193497282023559075, −9.319836877693985146454983831889, −8.487362310089932875725245890518, −8.128112844371914322565103432782, −6.88333188778539109788281407668, −5.21736118340404029860476845697, −3.86946173341877953391778426156, −3.03075064964641000905748004614, −1.39981815650734797040103600553, 0.26257680604016125587411747956, 2.14873534536635771953439758402, 3.78982754694596639733502080474, 5.64545962452296862488832142177, 6.64333523859920131465325409606, 7.05966738438608329908088049558, 8.035759949381033307570404856984, 8.584372158599094643839534553466, 9.631516055630741816645699107945, 10.41333374254406230617932267006

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