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Properties

Label 2-1503-1503.166-c0-0-8
Degree $2$
Conductor $1503$
Sign $-0.916 + 0.400i$
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  + (−0.580 − 1.00i)2-s + (0.928 − 0.371i)3-s + (−0.172 + 0.299i)4-s + (−0.911 − 0.717i)6-s + (−0.0475 − 0.0824i)7-s − 0.758·8-s + (0.723 − 0.690i)9-s + (−0.981 − 1.70i)11-s + (−0.0492 + 0.342i)12-s + (−0.0552 + 0.0956i)14-s + (0.613 + 1.06i)16-s + (−1.11 − 0.326i)18-s + 0.471·19-s + (−0.0748 − 0.0588i)21-s + (−1.13 + 1.97i)22-s + ⋯
L(s)  = 1  + (−0.580 − 1.00i)2-s + (0.928 − 0.371i)3-s + (−0.172 + 0.299i)4-s + (−0.911 − 0.717i)6-s + (−0.0475 − 0.0824i)7-s − 0.758·8-s + (0.723 − 0.690i)9-s + (−0.981 − 1.70i)11-s + (−0.0492 + 0.342i)12-s + (−0.0552 + 0.0956i)14-s + (0.613 + 1.06i)16-s + (−1.11 − 0.326i)18-s + 0.471·19-s + (−0.0748 − 0.0588i)21-s + (−1.13 + 1.97i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1503\)    =    \(3^{2} \cdot 167\)
Sign: $-0.916 + 0.400i$
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: \((2,\ 1503,\ (\ :0),\ -0.916 + 0.400i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.008216612\)
\(L(\frac12)\) \(\approx\) \(1.008216612\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.928 + 0.371i)T \)
167 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.580 + 1.00i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.0475 + 0.0824i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.981 + 1.70i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 0.471T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.888 - 1.53i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.580 - 1.00i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.723 + 1.25i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.142 - 0.246i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 - 1.85T + T^{2} \)
97 \( 1 + (-0.654 - 1.13i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.274363891237584008791039069791, −8.563564030039721298827476327857, −8.207968535472719213936354762359, −7.06974731561348834823590085644, −6.14998759444219713719162575681, −5.16075924223843340684675374046, −3.53878890409344785045347800799, −3.09810141255693169531456332801, −2.12209605220003440653709383077, −0.878050014798597375235898463768, 2.09485811408538418310023457044, 2.98639689500156224554563984082, 4.22321712001306769442901006760, 5.09871575293362976610524931559, 6.12776515317865172302907264266, 7.25080531230268214974785673561, 7.64820704088815730627752991687, 8.184925085904707803333311627727, 9.276269390737736795900988565603, 9.641315385175069127490676263918

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