Properties

Label 2-531-531.121-c1-0-12
Degree $2$
Conductor $531$
Sign $0.183 + 0.982i$
Analytic cond. $4.24005$
Root an. cond. $2.05913$
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  + (−1.89 − 1.33i)2-s + (−0.834 − 1.51i)3-s + (1.14 + 3.19i)4-s + (−2.58 + 1.42i)5-s + (−0.447 + 3.99i)6-s + (−2.47 + 1.25i)7-s + (0.866 − 3.12i)8-s + (−1.60 + 2.53i)9-s + (6.81 + 0.741i)10-s + (−4.76 + 1.79i)11-s + (3.90 − 4.39i)12-s + (−0.187 − 1.47i)13-s + (6.38 + 0.929i)14-s + (4.32 + 2.72i)15-s + (−0.566 + 0.464i)16-s + (−0.328 − 6.05i)17-s + ⋯
L(s)  = 1  + (−1.34 − 0.946i)2-s + (−0.481 − 0.876i)3-s + (0.570 + 1.59i)4-s + (−1.15 + 0.639i)5-s + (−0.182 + 1.63i)6-s + (−0.937 + 0.475i)7-s + (0.306 − 1.10i)8-s + (−0.536 + 0.844i)9-s + (2.15 + 0.234i)10-s + (−1.43 + 0.542i)11-s + (1.12 − 1.26i)12-s + (−0.0519 − 0.408i)13-s + (1.70 + 0.248i)14-s + (1.11 + 0.704i)15-s + (−0.141 + 0.116i)16-s + (−0.0795 − 1.46i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.183 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.183 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $0.183 + 0.982i$
Analytic conductor: \(4.24005\)
Root analytic conductor: \(2.05913\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{531} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :1/2),\ 0.183 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.175214 - 0.145509i\)
\(L(\frac12)\) \(\approx\) \(0.175214 - 0.145509i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.834 + 1.51i)T \)
59 \( 1 + (2.08 - 7.39i)T \)
good2 \( 1 + (1.89 + 1.33i)T + (0.672 + 1.88i)T^{2} \)
5 \( 1 + (2.58 - 1.42i)T + (2.65 - 4.23i)T^{2} \)
7 \( 1 + (2.47 - 1.25i)T + (4.13 - 5.64i)T^{2} \)
11 \( 1 + (4.76 - 1.79i)T + (8.25 - 7.27i)T^{2} \)
13 \( 1 + (0.187 + 1.47i)T + (-12.5 + 3.25i)T^{2} \)
17 \( 1 + (0.328 + 6.05i)T + (-16.9 + 1.83i)T^{2} \)
19 \( 1 + (0.659 - 0.624i)T + (1.02 - 18.9i)T^{2} \)
23 \( 1 + (-1.11 + 3.51i)T + (-18.8 - 13.2i)T^{2} \)
29 \( 1 + (-0.347 - 3.83i)T + (-28.5 + 5.20i)T^{2} \)
31 \( 1 + (-9.87 + 2.93i)T + (25.9 - 16.9i)T^{2} \)
37 \( 1 + (-0.793 - 2.85i)T + (-31.7 + 19.0i)T^{2} \)
41 \( 1 + (-0.985 + 1.07i)T + (-3.69 - 40.8i)T^{2} \)
43 \( 1 + (2.69 - 2.20i)T + (8.48 - 42.1i)T^{2} \)
47 \( 1 + (1.36 + 0.754i)T + (24.9 + 39.8i)T^{2} \)
53 \( 1 + (-8.72 + 0.948i)T + (51.7 - 11.3i)T^{2} \)
61 \( 1 + (8.01 + 5.64i)T + (20.5 + 57.4i)T^{2} \)
67 \( 1 + (4.26 + 4.34i)T + (-1.20 + 66.9i)T^{2} \)
71 \( 1 + (3.75 - 2.25i)T + (33.2 - 62.7i)T^{2} \)
73 \( 1 + (-3.42 - 8.58i)T + (-52.9 + 50.2i)T^{2} \)
79 \( 1 + (8.34 + 7.34i)T + (9.95 + 78.3i)T^{2} \)
83 \( 1 + (-14.0 - 1.01i)T + (82.1 + 11.9i)T^{2} \)
89 \( 1 + (-11.7 - 5.43i)T + (57.6 + 67.8i)T^{2} \)
97 \( 1 + (-18.2 + 2.64i)T + (92.9 - 27.6i)T^{2} \)
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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

−10.59078044746216778878381390097, −10.05839853061664919033101631088, −8.865808271908439318101427285295, −7.87472274734053638616301828136, −7.46655505051258377348625428892, −6.48179506706146207597913495662, −4.98875054330227645521300045317, −2.98614206160992074682846430618, −2.56681771095959914549851856157, −0.48069445746691399761269828745, 0.53048026289703262186188663751, 3.43149875663059492017346567772, 4.51599260021202134930407110006, 5.75860624614989484656442581170, 6.57823452701822828057702497511, 7.73307716185138502798584557260, 8.361147486238403657788983288567, 9.116687264980933968630056869951, 10.17403105059644247853932797101, 10.51976169682404583087985761697

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