Properties

Label 2-4011-1.1-c1-0-83
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  − 0.223·2-s + 3-s − 1.95·4-s + 1.64·5-s − 0.223·6-s − 7-s + 0.882·8-s + 9-s − 0.367·10-s + 4.86·11-s − 1.95·12-s + 4.77·13-s + 0.223·14-s + 1.64·15-s + 3.70·16-s + 2.25·17-s − 0.223·18-s − 1.51·19-s − 3.20·20-s − 21-s − 1.08·22-s − 1.56·23-s + 0.882·24-s − 2.29·25-s − 1.06·26-s + 27-s + 1.95·28-s + ⋯
L(s)  = 1  − 0.157·2-s + 0.577·3-s − 0.975·4-s + 0.735·5-s − 0.0911·6-s − 0.377·7-s + 0.311·8-s + 0.333·9-s − 0.116·10-s + 1.46·11-s − 0.562·12-s + 1.32·13-s + 0.0596·14-s + 0.424·15-s + 0.925·16-s + 0.546·17-s − 0.0526·18-s − 0.348·19-s − 0.717·20-s − 0.218·21-s − 0.231·22-s − 0.326·23-s + 0.180·24-s − 0.458·25-s − 0.209·26-s + 0.192·27-s + 0.368·28-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

\(L(1)\) \(\approx\) \(2.354661252\)
\(L(\frac12)\) \(\approx\) \(2.354661252\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.223T + 2T^{2} \)
5 \( 1 - 1.64T + 5T^{2} \)
11 \( 1 - 4.86T + 11T^{2} \)
13 \( 1 - 4.77T + 13T^{2} \)
17 \( 1 - 2.25T + 17T^{2} \)
19 \( 1 + 1.51T + 19T^{2} \)
23 \( 1 + 1.56T + 23T^{2} \)
29 \( 1 - 5.30T + 29T^{2} \)
31 \( 1 + 5.90T + 31T^{2} \)
37 \( 1 - 9.20T + 37T^{2} \)
41 \( 1 + 6.69T + 41T^{2} \)
43 \( 1 - 9.36T + 43T^{2} \)
47 \( 1 + 2.53T + 47T^{2} \)
53 \( 1 - 6.61T + 53T^{2} \)
59 \( 1 + 6.16T + 59T^{2} \)
61 \( 1 - 7.37T + 61T^{2} \)
67 \( 1 + 0.551T + 67T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 - 3.39T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 + 8.33T + 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 - 5.63T + 97T^{2} \)
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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

−8.714838991534681179881771121425, −7.940684038978567020267581769340, −6.98486856164504786296103009608, −6.08967710290182559440176565141, −5.69878342358698736982025675785, −4.38567925533801212123055943299, −3.91055455486864352851905214548, −3.11398702019906515215469847350, −1.77405194916622801595493696954, −0.965671192613006786368568539119, 0.965671192613006786368568539119, 1.77405194916622801595493696954, 3.11398702019906515215469847350, 3.91055455486864352851905214548, 4.38567925533801212123055943299, 5.69878342358698736982025675785, 6.08967710290182559440176565141, 6.98486856164504786296103009608, 7.940684038978567020267581769340, 8.714838991534681179881771121425

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