Properties

Label 2-4031-1.1-c1-0-207
Degree $2$
Conductor $4031$
Sign $-1$
Analytic cond. $32.1876$
Root an. cond. $5.67342$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.41·2-s − 2.29·3-s + 3.85·4-s + 1.32·5-s + 5.55·6-s + 0.993·7-s − 4.49·8-s + 2.26·9-s − 3.20·10-s + 2.10·11-s − 8.85·12-s + 3.07·13-s − 2.40·14-s − 3.03·15-s + 3.15·16-s + 6.60·17-s − 5.48·18-s − 0.0631·19-s + 5.10·20-s − 2.28·21-s − 5.09·22-s − 5.94·23-s + 10.3·24-s − 3.25·25-s − 7.45·26-s + 1.67·27-s + 3.83·28-s + ⋯
L(s)  = 1  − 1.71·2-s − 1.32·3-s + 1.92·4-s + 0.591·5-s + 2.26·6-s + 0.375·7-s − 1.58·8-s + 0.756·9-s − 1.01·10-s + 0.634·11-s − 2.55·12-s + 0.853·13-s − 0.642·14-s − 0.783·15-s + 0.789·16-s + 1.60·17-s − 1.29·18-s − 0.0144·19-s + 1.14·20-s − 0.497·21-s − 1.08·22-s − 1.23·23-s + 2.10·24-s − 0.650·25-s − 1.46·26-s + 0.323·27-s + 0.724·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $-1$
Analytic conductor: \(32.1876\)
Root analytic conductor: \(5.67342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4031,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + T \)
139 \( 1 + T \)
good2 \( 1 + 2.41T + 2T^{2} \)
3 \( 1 + 2.29T + 3T^{2} \)
5 \( 1 - 1.32T + 5T^{2} \)
7 \( 1 - 0.993T + 7T^{2} \)
11 \( 1 - 2.10T + 11T^{2} \)
13 \( 1 - 3.07T + 13T^{2} \)
17 \( 1 - 6.60T + 17T^{2} \)
19 \( 1 + 0.0631T + 19T^{2} \)
23 \( 1 + 5.94T + 23T^{2} \)
31 \( 1 - 5.42T + 31T^{2} \)
37 \( 1 + 5.88T + 37T^{2} \)
41 \( 1 + 5.46T + 41T^{2} \)
43 \( 1 - 0.835T + 43T^{2} \)
47 \( 1 + 0.379T + 47T^{2} \)
53 \( 1 + 2.25T + 53T^{2} \)
59 \( 1 + 7.33T + 59T^{2} \)
61 \( 1 + 4.55T + 61T^{2} \)
67 \( 1 + 8.31T + 67T^{2} \)
71 \( 1 + 7.18T + 71T^{2} \)
73 \( 1 + 15.1T + 73T^{2} \)
79 \( 1 + 1.93T + 79T^{2} \)
83 \( 1 - 7.16T + 83T^{2} \)
89 \( 1 + 3.20T + 89T^{2} \)
97 \( 1 + 4.74T + 97T^{2} \)
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

−8.152861108817099477737352967783, −7.49818063653595384008946968921, −6.57287046680055360140732169754, −6.06121751067877184074261169833, −5.52744775065208792758188318875, −4.40718102656231644747114077895, −3.15808289014204550114328594725, −1.69350332097963047368611574394, −1.25401336107538663703940123637, 0, 1.25401336107538663703940123637, 1.69350332097963047368611574394, 3.15808289014204550114328594725, 4.40718102656231644747114077895, 5.52744775065208792758188318875, 6.06121751067877184074261169833, 6.57287046680055360140732169754, 7.49818063653595384008946968921, 8.152861108817099477737352967783

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