Properties

Label 2-4011-1.1-c1-0-108
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.44·2-s + 3-s + 0.0928·4-s + 2.29·5-s + 1.44·6-s + 7-s − 2.75·8-s + 9-s + 3.32·10-s − 1.55·11-s + 0.0928·12-s + 6.79·13-s + 1.44·14-s + 2.29·15-s − 4.17·16-s + 0.345·17-s + 1.44·18-s − 2.91·19-s + 0.213·20-s + 21-s − 2.24·22-s + 1.63·23-s − 2.75·24-s + 0.276·25-s + 9.82·26-s + 27-s + 0.0928·28-s + ⋯
L(s)  = 1  + 1.02·2-s + 0.577·3-s + 0.0464·4-s + 1.02·5-s + 0.590·6-s + 0.377·7-s − 0.975·8-s + 0.333·9-s + 1.05·10-s − 0.467·11-s + 0.0268·12-s + 1.88·13-s + 0.386·14-s + 0.593·15-s − 1.04·16-s + 0.0838·17-s + 0.340·18-s − 0.668·19-s + 0.0476·20-s + 0.218·21-s − 0.478·22-s + 0.340·23-s − 0.563·24-s + 0.0552·25-s + 1.92·26-s + 0.192·27-s + 0.0175·28-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

\(L(1)\) \(\approx\) \(4.809192933\)
\(L(\frac12)\) \(\approx\) \(4.809192933\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.44T + 2T^{2} \)
5 \( 1 - 2.29T + 5T^{2} \)
11 \( 1 + 1.55T + 11T^{2} \)
13 \( 1 - 6.79T + 13T^{2} \)
17 \( 1 - 0.345T + 17T^{2} \)
19 \( 1 + 2.91T + 19T^{2} \)
23 \( 1 - 1.63T + 23T^{2} \)
29 \( 1 + 0.433T + 29T^{2} \)
31 \( 1 - 4.80T + 31T^{2} \)
37 \( 1 - 10.0T + 37T^{2} \)
41 \( 1 - 3.90T + 41T^{2} \)
43 \( 1 - 1.20T + 43T^{2} \)
47 \( 1 + 0.954T + 47T^{2} \)
53 \( 1 - 8.65T + 53T^{2} \)
59 \( 1 - 8.57T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 + 4.10T + 67T^{2} \)
71 \( 1 - 1.62T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 + 6.27T + 83T^{2} \)
89 \( 1 + 8.34T + 89T^{2} \)
97 \( 1 + 2.81T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.534439609870125402838309180713, −7.83654032394565910386348653374, −6.64730286641231509181652173859, −5.99578147824496270485118491687, −5.57159535460392000877447580793, −4.53317491252451341412564818334, −3.97408520032292252815270022461, −3.01817068420627175647003112848, −2.27549011241920247807518820975, −1.13729999947655940400882125400, 1.13729999947655940400882125400, 2.27549011241920247807518820975, 3.01817068420627175647003112848, 3.97408520032292252815270022461, 4.53317491252451341412564818334, 5.57159535460392000877447580793, 5.99578147824496270485118491687, 6.64730286641231509181652173859, 7.83654032394565910386348653374, 8.534439609870125402838309180713

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