Properties

Label 2-4003-1.1-c1-0-63
Degree $2$
Conductor $4003$
Sign $1$
Analytic cond. $31.9641$
Root an. cond. $5.65368$
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  − 2.54·2-s − 0.919·3-s + 4.48·4-s − 0.00498·5-s + 2.34·6-s − 0.132·7-s − 6.33·8-s − 2.15·9-s + 0.0127·10-s + 5.60·11-s − 4.12·12-s + 4.28·13-s + 0.336·14-s + 0.00458·15-s + 7.16·16-s − 5.76·17-s + 5.48·18-s − 6.95·19-s − 0.0223·20-s + 0.121·21-s − 14.2·22-s + 6.39·23-s + 5.82·24-s − 4.99·25-s − 10.9·26-s + 4.73·27-s − 0.592·28-s + ⋯
L(s)  = 1  − 1.80·2-s − 0.530·3-s + 2.24·4-s − 0.00223·5-s + 0.956·6-s − 0.0499·7-s − 2.23·8-s − 0.718·9-s + 0.00401·10-s + 1.69·11-s − 1.19·12-s + 1.18·13-s + 0.0899·14-s + 0.00118·15-s + 1.79·16-s − 1.39·17-s + 1.29·18-s − 1.59·19-s − 0.00500·20-s + 0.0265·21-s − 3.04·22-s + 1.33·23-s + 1.18·24-s − 0.999·25-s − 2.13·26-s + 0.912·27-s − 0.112·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4003\)
Sign: $1$
Analytic conductor: \(31.9641\)
Root analytic conductor: \(5.65368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4003,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5716388904\)
\(L(\frac12)\) \(\approx\) \(0.5716388904\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1+O(T) \)
good2 \( 1 + 2.54T + 2T^{2} \)
3 \( 1 + 0.919T + 3T^{2} \)
5 \( 1 + 0.00498T + 5T^{2} \)
7 \( 1 + 0.132T + 7T^{2} \)
11 \( 1 - 5.60T + 11T^{2} \)
13 \( 1 - 4.28T + 13T^{2} \)
17 \( 1 + 5.76T + 17T^{2} \)
19 \( 1 + 6.95T + 19T^{2} \)
23 \( 1 - 6.39T + 23T^{2} \)
29 \( 1 - 1.11T + 29T^{2} \)
31 \( 1 - 6.59T + 31T^{2} \)
37 \( 1 - 3.65T + 37T^{2} \)
41 \( 1 + 3.61T + 41T^{2} \)
43 \( 1 - 1.54T + 43T^{2} \)
47 \( 1 + 0.423T + 47T^{2} \)
53 \( 1 + 0.698T + 53T^{2} \)
59 \( 1 - 0.427T + 59T^{2} \)
61 \( 1 + 8.82T + 61T^{2} \)
67 \( 1 - 14.5T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + 7.78T + 73T^{2} \)
79 \( 1 - 8.90T + 79T^{2} \)
83 \( 1 - 3.14T + 83T^{2} \)
89 \( 1 - 9.35T + 89T^{2} \)
97 \( 1 + 8.80T + 97T^{2} \)
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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

−8.556746422274107761309359199101, −8.114224051016028528269170663512, −6.84075057717665936129173583119, −6.42534908112973310668577522551, −6.14589287931067423632903501800, −4.68204228577925097727456769962, −3.69996735222961516174785712051, −2.51577517709618553326107695126, −1.56199249210938897424165674955, −0.60660800596873391670564453151, 0.60660800596873391670564453151, 1.56199249210938897424165674955, 2.51577517709618553326107695126, 3.69996735222961516174785712051, 4.68204228577925097727456769962, 6.14589287931067423632903501800, 6.42534908112973310668577522551, 6.84075057717665936129173583119, 8.114224051016028528269170663512, 8.556746422274107761309359199101

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