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Properties

Label 2-1503-1503.166-c0-0-6
Degree $2$
Conductor $1503$
Sign $0.296 - 0.954i$
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  + (0.327 + 0.566i)2-s + (0.723 + 0.690i)3-s + (0.286 − 0.495i)4-s + (−0.154 + 0.635i)6-s + (0.995 + 1.72i)7-s + 1.02·8-s + (0.0475 + 0.998i)9-s + (−0.928 − 1.60i)11-s + (0.548 − 0.161i)12-s + (−0.651 + 1.12i)14-s + (0.0502 + 0.0871i)16-s + (−0.550 + 0.353i)18-s − 1.77·19-s + (−0.469 + 1.93i)21-s + (0.607 − 1.05i)22-s + ⋯
L(s)  = 1  + (0.327 + 0.566i)2-s + (0.723 + 0.690i)3-s + (0.286 − 0.495i)4-s + (−0.154 + 0.635i)6-s + (0.995 + 1.72i)7-s + 1.02·8-s + (0.0475 + 0.998i)9-s + (−0.928 − 1.60i)11-s + (0.548 − 0.161i)12-s + (−0.651 + 1.12i)14-s + (0.0502 + 0.0871i)16-s + (−0.550 + 0.353i)18-s − 1.77·19-s + (−0.469 + 1.93i)21-s + (0.607 − 1.05i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1503\)    =    \(3^{2} \cdot 167\)
Sign: $0.296 - 0.954i$
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: \((2,\ 1503,\ (\ :0),\ 0.296 - 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.888703271\)
\(L(\frac12)\) \(\approx\) \(1.888703271\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.723 - 0.690i)T \)
167 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.327 - 0.566i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.995 - 1.72i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.928 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.77T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.580 + 1.00i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.327 + 0.566i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.0475 + 0.0824i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.959 - 1.66i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 - 1.44T + T^{2} \)
97 \( 1 + (-0.142 - 0.246i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.776891251859034949271317470209, −8.717029372329769191932925469159, −8.333475826800019458058152946603, −7.74327991469590657631102930350, −6.16980908553059436694228029925, −5.74930492637676787771057949666, −4.99860578291501726139762976204, −4.13056994900202833727031212673, −2.61276935123263120557580161446, −2.11404663261894957064802076386, 1.60730189065608048597452288240, 2.17530743654655514447522587235, 3.48962714469747787383168977554, 4.25016367852966944553850658454, 4.93332595360867198287604543653, 6.78640481296624759903634939502, 7.19189044198833888644555554489, 7.84450142073918388032552312741, 8.360383990719073776190515956770, 9.667340545696974472828314429431

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