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Properties

Label 2-1503-1503.166-c0-0-0
Degree $2$
Conductor $1503$
Sign $0.991 - 0.126i$
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  + (−0.981 − 1.70i)2-s + (0.235 + 0.971i)3-s + (−1.42 + 2.47i)4-s + (1.42 − 1.35i)6-s + (−0.580 − 1.00i)7-s + 3.64·8-s + (−0.888 + 0.458i)9-s + (0.786 + 1.36i)11-s + (−2.74 − 0.804i)12-s + (−1.13 + 1.97i)14-s + (−2.15 − 3.72i)16-s + (1.65 + 1.06i)18-s + 0.0951·19-s + (0.839 − 0.800i)21-s + (1.54 − 2.67i)22-s + ⋯
L(s)  = 1  + (−0.981 − 1.70i)2-s + (0.235 + 0.971i)3-s + (−1.42 + 2.47i)4-s + (1.42 − 1.35i)6-s + (−0.580 − 1.00i)7-s + 3.64·8-s + (−0.888 + 0.458i)9-s + (0.786 + 1.36i)11-s + (−2.74 − 0.804i)12-s + (−1.13 + 1.97i)14-s + (−2.15 − 3.72i)16-s + (1.65 + 1.06i)18-s + 0.0951·19-s + (0.839 − 0.800i)21-s + (1.54 − 2.67i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1503\)    =    \(3^{2} \cdot 167\)
Sign: $0.991 - 0.126i$
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: \((2,\ 1503,\ (\ :0),\ 0.991 - 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5470508199\)
\(L(\frac12)\) \(\approx\) \(0.5470508199\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.235 - 0.971i)T \)
167 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.981 + 1.70i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.580 + 1.00i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.786 - 1.36i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 0.0951T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.995 - 1.72i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.981 - 1.70i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.888 - 1.53i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 - 0.471T + T^{2} \)
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

−9.892229827749425205142461426268, −9.158232319744926603508144707091, −8.660758122615566959815818976615, −7.56800468131095395971838886155, −6.86495070746636553882468390949, −4.91300470244321027031921942915, −4.18443228507547817130579923364, −3.57132212321901188681406661052, −2.65009240286192305592279733163, −1.37880262668379416661136073264, 0.65698921081762489730133640662, 2.14396614264764320861934669105, 3.76951954298451061392666119158, 5.39745113940026782130971969296, 6.02955094468542140066675140982, 6.35687312855132268384747381469, 7.31885071287591360388798003603, 8.063353235136471988751403852460, 8.677133821625724825571154469054, 9.234458619707427028752092667575

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