Properties

Label 2-4011-1.1-c1-0-84
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  − 1.47·2-s + 3-s + 0.186·4-s + 3.98·5-s − 1.47·6-s − 7-s + 2.68·8-s + 9-s − 5.89·10-s + 4.84·11-s + 0.186·12-s + 1.76·13-s + 1.47·14-s + 3.98·15-s − 4.33·16-s − 4.24·17-s − 1.47·18-s − 3.33·19-s + 0.742·20-s − 21-s − 7.15·22-s − 6.65·23-s + 2.68·24-s + 10.9·25-s − 2.60·26-s + 27-s − 0.186·28-s + ⋯
L(s)  = 1  − 1.04·2-s + 0.577·3-s + 0.0930·4-s + 1.78·5-s − 0.603·6-s − 0.377·7-s + 0.948·8-s + 0.333·9-s − 1.86·10-s + 1.46·11-s + 0.0537·12-s + 0.489·13-s + 0.395·14-s + 1.03·15-s − 1.08·16-s − 1.02·17-s − 0.348·18-s − 0.764·19-s + 0.166·20-s − 0.218·21-s − 1.52·22-s − 1.38·23-s + 0.547·24-s + 2.18·25-s − 0.511·26-s + 0.192·27-s − 0.0351·28-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

\(L(1)\) \(\approx\) \(1.932555595\)
\(L(\frac12)\) \(\approx\) \(1.932555595\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.47T + 2T^{2} \)
5 \( 1 - 3.98T + 5T^{2} \)
11 \( 1 - 4.84T + 11T^{2} \)
13 \( 1 - 1.76T + 13T^{2} \)
17 \( 1 + 4.24T + 17T^{2} \)
19 \( 1 + 3.33T + 19T^{2} \)
23 \( 1 + 6.65T + 23T^{2} \)
29 \( 1 + 5.64T + 29T^{2} \)
31 \( 1 - 9.97T + 31T^{2} \)
37 \( 1 - 8.75T + 37T^{2} \)
41 \( 1 - 6.47T + 41T^{2} \)
43 \( 1 + 8.81T + 43T^{2} \)
47 \( 1 - 8.40T + 47T^{2} \)
53 \( 1 - 0.147T + 53T^{2} \)
59 \( 1 - 2.19T + 59T^{2} \)
61 \( 1 - 15.1T + 61T^{2} \)
67 \( 1 + 8.76T + 67T^{2} \)
71 \( 1 - 7.02T + 71T^{2} \)
73 \( 1 - 1.50T + 73T^{2} \)
79 \( 1 + 2.89T + 79T^{2} \)
83 \( 1 - 2.65T + 83T^{2} \)
89 \( 1 - 8.87T + 89T^{2} \)
97 \( 1 + 4.53T + 97T^{2} \)
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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

−8.683859254490084180940187356938, −8.066084524731464672092120228818, −6.88491246418661013103214082103, −6.41843503311389914296909270258, −5.79012388301234768554680713767, −4.50615658007192653423965340402, −3.91231710319161383018991420094, −2.45427590962314140964571828063, −1.88305488504891261927107272403, −0.967248624813328352898398324584, 0.967248624813328352898398324584, 1.88305488504891261927107272403, 2.45427590962314140964571828063, 3.91231710319161383018991420094, 4.50615658007192653423965340402, 5.79012388301234768554680713767, 6.41843503311389914296909270258, 6.88491246418661013103214082103, 8.066084524731464672092120228818, 8.683859254490084180940187356938

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