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Properties

Label 2-1503-1503.1168-c0-0-2
Degree $2$
Conductor $1503$
Sign $0.527 - 0.849i$
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  + (−0.0475 + 0.0824i)2-s + (−0.327 − 0.945i)3-s + (0.495 + 0.858i)4-s + (0.0934 + 0.0180i)6-s + (−0.235 + 0.408i)7-s − 0.189·8-s + (−0.786 + 0.618i)9-s + (−0.580 + 1.00i)11-s + (0.648 − 0.748i)12-s + (−0.0224 − 0.0388i)14-s + (−0.486 + 0.842i)16-s + (−0.0135 − 0.0941i)18-s + 1.85·19-s + (0.462 + 0.0892i)21-s + (−0.0552 − 0.0956i)22-s + ⋯
L(s)  = 1  + (−0.0475 + 0.0824i)2-s + (−0.327 − 0.945i)3-s + (0.495 + 0.858i)4-s + (0.0934 + 0.0180i)6-s + (−0.235 + 0.408i)7-s − 0.189·8-s + (−0.786 + 0.618i)9-s + (−0.580 + 1.00i)11-s + (0.648 − 0.748i)12-s + (−0.0224 − 0.0388i)14-s + (−0.486 + 0.842i)16-s + (−0.0135 − 0.0941i)18-s + 1.85·19-s + (0.462 + 0.0892i)21-s + (−0.0552 − 0.0956i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1503\)    =    \(3^{2} \cdot 167\)
Sign: $0.527 - 0.849i$
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: \((2,\ 1503,\ (\ :0),\ 0.527 - 0.849i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9237447758\)
\(L(\frac12)\) \(\approx\) \(0.9237447758\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.327 + 0.945i)T \)
167 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.0475 - 0.0824i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.235 - 0.408i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.580 - 1.00i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.85T + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.723 - 1.25i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.0475 + 0.0824i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.786 + 1.36i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.654 + 1.13i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + 0.654T + T^{2} \)
97 \( 1 + (0.415 - 0.719i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.684231426706316202179851168516, −8.877002324143706010466093944076, −7.903453665043293973899376804798, −7.33087060768339439796683807185, −6.90186733203946353065220529320, −5.73861048280669678447128277062, −5.09446213001530818737324948691, −3.57706330562578483059731271936, −2.67574144539342170127944114297, −1.70323546421535368409084072573, 0.78719637920725477006538219327, 2.58864846093156581215128439564, 3.50874300026328020722777123394, 4.58453146629761587265594166144, 5.69321763735759650553669083045, 5.84253429645758106248067764196, 7.03742656369315947048327472345, 7.969521739684597646540598892188, 9.031748572958280561187791634349, 9.827244851174187680414283353588

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