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Properties

Label 2-1503-1503.166-c0-0-10
Degree $2$
Conductor $1503$
Sign $-0.987 - 0.158i$
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  + (−0.235 − 0.408i)2-s + (−0.995 + 0.0950i)3-s + (0.388 − 0.673i)4-s + (0.273 + 0.384i)6-s + (−0.928 − 1.60i)7-s − 0.838·8-s + (0.981 − 0.189i)9-s + (−0.0475 − 0.0824i)11-s + (−0.323 + 0.707i)12-s + (−0.437 + 0.758i)14-s + (−0.191 − 0.331i)16-s + (−0.308 − 0.356i)18-s − 0.654·19-s + (1.07 + 1.51i)21-s + (−0.0224 + 0.0388i)22-s + ⋯
L(s)  = 1  + (−0.235 − 0.408i)2-s + (−0.995 + 0.0950i)3-s + (0.388 − 0.673i)4-s + (0.273 + 0.384i)6-s + (−0.928 − 1.60i)7-s − 0.838·8-s + (0.981 − 0.189i)9-s + (−0.0475 − 0.0824i)11-s + (−0.323 + 0.707i)12-s + (−0.437 + 0.758i)14-s + (−0.191 − 0.331i)16-s + (−0.308 − 0.356i)18-s − 0.654·19-s + (1.07 + 1.51i)21-s + (−0.0224 + 0.0388i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1503\)    =    \(3^{2} \cdot 167\)
Sign: $-0.987 - 0.158i$
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: \((2,\ 1503,\ (\ :0),\ -0.987 - 0.158i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4170323316\)
\(L(\frac12)\) \(\approx\) \(0.4170323316\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.995 - 0.0950i)T \)
167 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.235 + 0.408i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.928 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.0475 + 0.0824i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 0.654T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.786 - 1.36i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.235 - 0.408i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.981 + 1.70i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.415 + 0.719i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + 1.99T + T^{2} \)
97 \( 1 + (0.841 + 1.45i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.844963352690163031318864934580, −8.617101296320350921413214620956, −7.28120318877735928311047281643, −6.71493246995570470423825776459, −6.18003870882079163194440085770, −5.11866970864113728057673379069, −4.17263562529691466201648328318, −3.19033554141767506698427038413, −1.59288647317470438617910280532, −0.39511257561634008756486624954, 2.12411195596493788798965879338, 3.07754476697578497134935146306, 4.29783289794147097073578854617, 5.54850558410793949518702137403, 6.12374183155478771153085210906, 6.64981043939901317510057563330, 7.63059441980667841839442391704, 8.429518974393519066271965142143, 9.320138739472767306279796621443, 9.878178652276357524631395280145

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