Properties

Label 2-4011-1.1-c1-0-146
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  + 2.68·2-s + 3-s + 5.20·4-s − 1.64·5-s + 2.68·6-s + 7-s + 8.60·8-s + 9-s − 4.42·10-s − 1.55·11-s + 5.20·12-s + 1.36·13-s + 2.68·14-s − 1.64·15-s + 12.6·16-s + 3.98·17-s + 2.68·18-s + 4.09·19-s − 8.57·20-s + 21-s − 4.17·22-s − 1.17·23-s + 8.60·24-s − 2.28·25-s + 3.65·26-s + 27-s + 5.20·28-s + ⋯
L(s)  = 1  + 1.89·2-s + 0.577·3-s + 2.60·4-s − 0.736·5-s + 1.09·6-s + 0.377·7-s + 3.04·8-s + 0.333·9-s − 1.39·10-s − 0.469·11-s + 1.50·12-s + 0.378·13-s + 0.717·14-s − 0.425·15-s + 3.17·16-s + 0.965·17-s + 0.632·18-s + 0.938·19-s − 1.91·20-s + 0.218·21-s − 0.890·22-s − 0.245·23-s + 1.75·24-s − 0.457·25-s + 0.717·26-s + 0.192·27-s + 0.983·28-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

\(L(1)\) \(\approx\) \(8.027555863\)
\(L(\frac12)\) \(\approx\) \(8.027555863\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.68T + 2T^{2} \)
5 \( 1 + 1.64T + 5T^{2} \)
11 \( 1 + 1.55T + 11T^{2} \)
13 \( 1 - 1.36T + 13T^{2} \)
17 \( 1 - 3.98T + 17T^{2} \)
19 \( 1 - 4.09T + 19T^{2} \)
23 \( 1 + 1.17T + 23T^{2} \)
29 \( 1 + 1.85T + 29T^{2} \)
31 \( 1 - 6.08T + 31T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
41 \( 1 + 1.99T + 41T^{2} \)
43 \( 1 - 3.05T + 43T^{2} \)
47 \( 1 - 7.18T + 47T^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
59 \( 1 + 5.41T + 59T^{2} \)
61 \( 1 + 12.9T + 61T^{2} \)
67 \( 1 + 1.92T + 67T^{2} \)
71 \( 1 + 1.64T + 71T^{2} \)
73 \( 1 + 6.10T + 73T^{2} \)
79 \( 1 + 0.436T + 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 - 6.60T + 89T^{2} \)
97 \( 1 + 13.1T + 97T^{2} \)
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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

−7.980730158002649454444493828420, −7.61608800656674429947715781679, −6.95414175719927916018046653347, −5.93685962463802413018595657879, −5.34013072137048538840187481920, −4.57258238294157857313316517085, −3.79521231858957030677497860693, −3.28921034128379239376945520627, −2.45014924915856830625376957278, −1.36352369775640794316267247281, 1.36352369775640794316267247281, 2.45014924915856830625376957278, 3.28921034128379239376945520627, 3.79521231858957030677497860693, 4.57258238294157857313316517085, 5.34013072137048538840187481920, 5.93685962463802413018595657879, 6.95414175719927916018046653347, 7.61608800656674429947715781679, 7.980730158002649454444493828420

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