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Properties

Label 2-1503-1503.166-c0-0-4
Degree $2$
Conductor $1503$
Sign $-0.745 - 0.666i$
Analytic cond. $0.750094$
Root an. cond. $0.866080$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.888 + 1.53i)2-s + (0.981 + 0.189i)3-s + (−1.08 + 1.87i)4-s + (0.581 + 1.67i)6-s + (−0.723 − 1.25i)7-s − 2.06·8-s + (0.928 + 0.371i)9-s + (0.995 + 1.72i)11-s + (−1.41 + 1.63i)12-s + (1.28 − 2.22i)14-s + (−0.752 − 1.30i)16-s + (0.252 + 1.75i)18-s − 1.57·19-s + (−0.473 − 1.36i)21-s + (−1.76 + 3.06i)22-s + ⋯
L(s)  = 1  + (0.888 + 1.53i)2-s + (0.981 + 0.189i)3-s + (−1.08 + 1.87i)4-s + (0.581 + 1.67i)6-s + (−0.723 − 1.25i)7-s − 2.06·8-s + (0.928 + 0.371i)9-s + (0.995 + 1.72i)11-s + (−1.41 + 1.63i)12-s + (1.28 − 2.22i)14-s + (−0.752 − 1.30i)16-s + (0.252 + 1.75i)18-s − 1.57·19-s + (−0.473 − 1.36i)21-s + (−1.76 + 3.06i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1503\)    =    \(3^{2} \cdot 167\)
Sign: $-0.745 - 0.666i$
Analytic conductor: \(0.750094\)
Root analytic conductor: \(0.866080\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1503} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1503,\ (\ :0),\ -0.745 - 0.666i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.133805630\)
\(L(\frac12)\) \(\approx\) \(2.133805630\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.981 - 0.189i)T \)
167 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.888 - 1.53i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.723 + 1.25i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.995 - 1.72i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.57T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.235 + 0.408i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.888 + 1.53i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.928 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.654 - 1.13i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - 1.96T + T^{2} \)
97 \( 1 + (0.415 + 0.719i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.859721941974813225214304657678, −8.929689356386849612839283712216, −8.061822542821388197085096226030, −7.39319377974358733860210346130, −6.76192198823090940850041704614, −6.26900408569863901701429905728, −4.69704521736147416746683989812, −4.15449323192951385483035394588, −3.75725185663301016879089205180, −2.19966396062171800131086371644, 1.40346178543609988657623059464, 2.46840208135396953206804360455, 3.25360354841189909550055339591, 3.72928955205501564128610999098, 4.89450789750331213951063481204, 6.02631599711310672672771985478, 6.52548472657648485861816629343, 8.218023323144897711123791482485, 8.965450223596379125516432862959, 9.279828369476016705503132959444

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