Properties

Label 2-8003-1.1-c1-0-59
Degree $2$
Conductor $8003$
Sign $1$
Analytic cond. $63.9042$
Root an. cond. $7.99401$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.74·2-s + 0.238·3-s + 5.55·4-s − 0.412·5-s − 0.655·6-s − 2.19·7-s − 9.76·8-s − 2.94·9-s + 1.13·10-s − 2.64·11-s + 1.32·12-s + 3.93·13-s + 6.02·14-s − 0.0983·15-s + 15.7·16-s − 5.33·17-s + 8.08·18-s − 1.03·19-s − 2.29·20-s − 0.522·21-s + 7.25·22-s + 1.82·23-s − 2.32·24-s − 4.82·25-s − 10.8·26-s − 1.41·27-s − 12.1·28-s + ⋯
L(s)  = 1  − 1.94·2-s + 0.137·3-s + 2.77·4-s − 0.184·5-s − 0.267·6-s − 0.828·7-s − 3.45·8-s − 0.981·9-s + 0.358·10-s − 0.796·11-s + 0.382·12-s + 1.09·13-s + 1.60·14-s − 0.0253·15-s + 3.93·16-s − 1.29·17-s + 1.90·18-s − 0.237·19-s − 0.512·20-s − 0.114·21-s + 1.54·22-s + 0.381·23-s − 0.475·24-s − 0.965·25-s − 2.12·26-s − 0.272·27-s − 2.30·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8003\)    =    \(53 \cdot 151\)
Sign: $1$
Analytic conductor: \(63.9042\)
Root analytic conductor: \(7.99401\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8003,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2120987215\)
\(L(\frac12)\) \(\approx\) \(0.2120987215\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad53 \( 1 + T \)
151 \( 1 - T \)
good2 \( 1 + 2.74T + 2T^{2} \)
3 \( 1 - 0.238T + 3T^{2} \)
5 \( 1 + 0.412T + 5T^{2} \)
7 \( 1 + 2.19T + 7T^{2} \)
11 \( 1 + 2.64T + 11T^{2} \)
13 \( 1 - 3.93T + 13T^{2} \)
17 \( 1 + 5.33T + 17T^{2} \)
19 \( 1 + 1.03T + 19T^{2} \)
23 \( 1 - 1.82T + 23T^{2} \)
29 \( 1 - 0.842T + 29T^{2} \)
31 \( 1 - 1.06T + 31T^{2} \)
37 \( 1 - 8.32T + 37T^{2} \)
41 \( 1 + 3.45T + 41T^{2} \)
43 \( 1 - 7.74T + 43T^{2} \)
47 \( 1 + 12.4T + 47T^{2} \)
59 \( 1 + 1.21T + 59T^{2} \)
61 \( 1 + 5.39T + 61T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 - 1.48T + 71T^{2} \)
73 \( 1 + 16.8T + 73T^{2} \)
79 \( 1 + 1.41T + 79T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 + 4.55T + 89T^{2} \)
97 \( 1 + 3.08T + 97T^{2} \)
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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

−8.168590316376121013674599763275, −7.40668977714141858217354255464, −6.46705689779012406439083069260, −6.28599748962958721444077278940, −5.42472293159041942392918582305, −3.99868183845599482612441370199, −2.96798577385461639155200746599, −2.58609856214543965447553127061, −1.53790075781868685541995195830, −0.29719725746046137883314215359, 0.29719725746046137883314215359, 1.53790075781868685541995195830, 2.58609856214543965447553127061, 2.96798577385461639155200746599, 3.99868183845599482612441370199, 5.42472293159041942392918582305, 6.28599748962958721444077278940, 6.46705689779012406439083069260, 7.40668977714141858217354255464, 8.168590316376121013674599763275

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