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Properties

Label 2-1503-1503.1168-c0-0-0
Degree $2$
Conductor $1503$
Sign $-0.266 + 0.963i$
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  + (−0.928 + 1.60i)2-s + (−0.888 + 0.458i)3-s + (−1.22 − 2.11i)4-s + (0.0883 − 1.85i)6-s + (0.327 − 0.566i)7-s + 2.68·8-s + (0.580 − 0.814i)9-s + (−0.235 + 0.408i)11-s + (2.05 + 1.32i)12-s + (0.607 + 1.05i)14-s + (−1.27 + 2.20i)16-s + (0.771 + 1.68i)18-s − 1.99·19-s + (−0.0311 + 0.653i)21-s + (−0.437 − 0.758i)22-s + ⋯
L(s)  = 1  + (−0.928 + 1.60i)2-s + (−0.888 + 0.458i)3-s + (−1.22 − 2.11i)4-s + (0.0883 − 1.85i)6-s + (0.327 − 0.566i)7-s + 2.68·8-s + (0.580 − 0.814i)9-s + (−0.235 + 0.408i)11-s + (2.05 + 1.32i)12-s + (0.607 + 1.05i)14-s + (−1.27 + 2.20i)16-s + (0.771 + 1.68i)18-s − 1.99·19-s + (−0.0311 + 0.653i)21-s + (−0.437 − 0.758i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1503\)    =    \(3^{2} \cdot 167\)
Sign: $-0.266 + 0.963i$
Analytic conductor: \(0.750094\)
Root analytic conductor: \(0.866080\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1503} (1168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1503,\ (\ :0),\ -0.266 + 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1253778833\)
\(L(\frac12)\) \(\approx\) \(0.1253778833\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.888 - 0.458i)T \)
167 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.928 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.327 + 0.566i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.235 - 0.408i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.99T + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.981 - 1.70i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.928 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.580 - 1.00i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.841 - 1.45i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + 1.77T + T^{2} \)
97 \( 1 + (-0.959 + 1.66i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.02692223289423447493289023617, −9.308280512937622917387284521720, −8.645387540627023772437080970033, −7.55351933157942514000813855909, −7.16685611561122129356978429261, −6.21366454946217817728791747885, −5.62976755880834242752151497056, −4.72410631169925698368083656065, −4.01578319311297618825109739275, −1.56395743915187067037306007022, 0.15495342712190889598668668894, 1.78216389281521211174669203058, 2.36841707744608454402740771431, 3.77105568539262172448070454077, 4.67698695073534059637298055464, 5.77380108808498095052950776462, 6.78593341005534625931141925514, 7.980005239859683402807383612146, 8.381273395991561464709013554629, 9.262761363806102738373235913357

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