Properties

Label 2-4011-1.1-c1-0-22
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  − 0.254·2-s − 3-s − 1.93·4-s − 1.93·5-s + 0.254·6-s − 7-s + 8-s + 9-s + 0.491·10-s + 3.87·11-s + 1.93·12-s + 5.68·13-s + 0.254·14-s + 1.93·15-s + 3.61·16-s − 5.69·17-s − 0.254·18-s + 2.31·19-s + 3.74·20-s + 21-s − 0.983·22-s − 3.50·23-s − 24-s − 1.25·25-s − 1.44·26-s − 27-s + 1.93·28-s + ⋯
L(s)  = 1  − 0.179·2-s − 0.577·3-s − 0.967·4-s − 0.865·5-s + 0.103·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.155·10-s + 1.16·11-s + 0.558·12-s + 1.57·13-s + 0.0679·14-s + 0.499·15-s + 0.904·16-s − 1.38·17-s − 0.0598·18-s + 0.531·19-s + 0.837·20-s + 0.218·21-s − 0.209·22-s − 0.731·23-s − 0.204·24-s − 0.250·25-s − 0.283·26-s − 0.192·27-s + 0.365·28-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

\(L(1)\) \(\approx\) \(0.7480942022\)
\(L(\frac12)\) \(\approx\) \(0.7480942022\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.254T + 2T^{2} \)
5 \( 1 + 1.93T + 5T^{2} \)
11 \( 1 - 3.87T + 11T^{2} \)
13 \( 1 - 5.68T + 13T^{2} \)
17 \( 1 + 5.69T + 17T^{2} \)
19 \( 1 - 2.31T + 19T^{2} \)
23 \( 1 + 3.50T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + 0.826T + 31T^{2} \)
37 \( 1 + 3.44T + 37T^{2} \)
41 \( 1 - 2.82T + 41T^{2} \)
43 \( 1 - 3.49T + 43T^{2} \)
47 \( 1 + 3.66T + 47T^{2} \)
53 \( 1 + 1.27T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 - 2.38T + 61T^{2} \)
67 \( 1 + 3.06T + 67T^{2} \)
71 \( 1 - 8.72T + 71T^{2} \)
73 \( 1 - 5.55T + 73T^{2} \)
79 \( 1 - 8.18T + 79T^{2} \)
83 \( 1 + 9.69T + 83T^{2} \)
89 \( 1 - 3.91T + 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−8.457166762869705212916886207855, −7.904057512422382858565046125059, −6.85785141330466255198531989748, −6.29717176706829382061657800519, −5.51337100763001135609865203259, −4.38505883248875814239641039751, −4.04899434164127530734691093792, −3.33567700651976668361911816105, −1.59274361085997758472205017177, −0.55881691903650565761123273360, 0.55881691903650565761123273360, 1.59274361085997758472205017177, 3.33567700651976668361911816105, 4.04899434164127530734691093792, 4.38505883248875814239641039751, 5.51337100763001135609865203259, 6.29717176706829382061657800519, 6.85785141330466255198531989748, 7.904057512422382858565046125059, 8.457166762869705212916886207855

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