Properties

Label 2-4011-1.1-c1-0-15
Degree $2$
Conductor $4011$
Sign $1$
Analytic cond. $32.0279$
Root an. cond. $5.65932$
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  − 1.86·2-s − 3-s + 1.46·4-s + 1.46·5-s + 1.86·6-s − 7-s + 8-s + 9-s − 2.72·10-s − 2.92·11-s − 1.46·12-s + 0.676·13-s + 1.86·14-s − 1.46·15-s − 4.78·16-s − 7.11·17-s − 1.86·18-s + 7.32·19-s + 2.13·20-s + 21-s + 5.44·22-s − 6.72·23-s − 24-s − 2.86·25-s − 1.25·26-s − 27-s − 1.46·28-s + ⋯
L(s)  = 1  − 1.31·2-s − 0.577·3-s + 0.731·4-s + 0.654·5-s + 0.759·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.860·10-s − 0.881·11-s − 0.422·12-s + 0.187·13-s + 0.497·14-s − 0.377·15-s − 1.19·16-s − 1.72·17-s − 0.438·18-s + 1.68·19-s + 0.478·20-s + 0.218·21-s + 1.16·22-s − 1.40·23-s − 0.204·24-s − 0.572·25-s − 0.246·26-s − 0.192·27-s − 0.276·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $1$
Analytic conductor: \(32.0279\)
Root analytic conductor: \(5.65932\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4011,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4774462211\)
\(L(\frac12)\) \(\approx\) \(0.4774462211\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + T \)
191 \( 1 - T \)
good2 \( 1 + 1.86T + 2T^{2} \)
5 \( 1 - 1.46T + 5T^{2} \)
11 \( 1 + 2.92T + 11T^{2} \)
13 \( 1 - 0.676T + 13T^{2} \)
17 \( 1 + 7.11T + 17T^{2} \)
19 \( 1 - 7.32T + 19T^{2} \)
23 \( 1 + 6.72T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + 9.04T + 31T^{2} \)
37 \( 1 + 3.25T + 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 - 0.278T + 43T^{2} \)
47 \( 1 - 7.76T + 47T^{2} \)
53 \( 1 + 9.30T + 53T^{2} \)
59 \( 1 - 4.71T + 59T^{2} \)
61 \( 1 - 10.7T + 61T^{2} \)
67 \( 1 + 6.46T + 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 - 5.74T + 73T^{2} \)
79 \( 1 - 6.39T + 79T^{2} \)
83 \( 1 + 11.1T + 83T^{2} \)
89 \( 1 + 5.90T + 89T^{2} \)
97 \( 1 - 17.5T + 97T^{2} \)
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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

−8.656806164389547171194062365515, −7.60460320017805967363235844058, −7.30855755687492464258846838049, −6.28249768223238983459308463717, −5.69128864789993793877536935360, −4.82429379188155609990241812197, −3.87244444709663126090545606047, −2.49199329038183555160502862064, −1.75441985654495871116973962895, −0.49059418597479321147403389026, 0.49059418597479321147403389026, 1.75441985654495871116973962895, 2.49199329038183555160502862064, 3.87244444709663126090545606047, 4.82429379188155609990241812197, 5.69128864789993793877536935360, 6.28249768223238983459308463717, 7.30855755687492464258846838049, 7.60460320017805967363235844058, 8.656806164389547171194062365515

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