Properties

Label 2-4014-1.1-c1-0-32
Degree $2$
Conductor $4014$
Sign $1$
Analytic cond. $32.0519$
Root an. cond. $5.66144$
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-s + 4-s + 3.52·5-s − 0.552·7-s − 8-s − 3.52·10-s − 4.67·11-s + 4.98·13-s + 0.552·14-s + 16-s + 5.42·17-s + 5.81·19-s + 3.52·20-s + 4.67·22-s − 1.00·23-s + 7.41·25-s − 4.98·26-s − 0.552·28-s − 1.30·29-s − 2.55·31-s − 32-s − 5.42·34-s − 1.94·35-s + 6.51·37-s − 5.81·38-s − 3.52·40-s + 1.43·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.57·5-s − 0.208·7-s − 0.353·8-s − 1.11·10-s − 1.41·11-s + 1.38·13-s + 0.147·14-s + 0.250·16-s + 1.31·17-s + 1.33·19-s + 0.787·20-s + 0.997·22-s − 0.210·23-s + 1.48·25-s − 0.977·26-s − 0.104·28-s − 0.241·29-s − 0.458·31-s − 0.176·32-s − 0.930·34-s − 0.328·35-s + 1.07·37-s − 0.943·38-s − 0.557·40-s + 0.224·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4014\)    =    \(2 \cdot 3^{2} \cdot 223\)
Sign: $1$
Analytic conductor: \(32.0519\)
Root analytic conductor: \(5.66144\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4014,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.981444404\)
\(L(\frac12)\) \(\approx\) \(1.981444404\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
223 \( 1 + T \)
good5 \( 1 - 3.52T + 5T^{2} \)
7 \( 1 + 0.552T + 7T^{2} \)
11 \( 1 + 4.67T + 11T^{2} \)
13 \( 1 - 4.98T + 13T^{2} \)
17 \( 1 - 5.42T + 17T^{2} \)
19 \( 1 - 5.81T + 19T^{2} \)
23 \( 1 + 1.00T + 23T^{2} \)
29 \( 1 + 1.30T + 29T^{2} \)
31 \( 1 + 2.55T + 31T^{2} \)
37 \( 1 - 6.51T + 37T^{2} \)
41 \( 1 - 1.43T + 41T^{2} \)
43 \( 1 + 2.12T + 43T^{2} \)
47 \( 1 + 9.21T + 47T^{2} \)
53 \( 1 - 11.7T + 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 14.7T + 67T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + 1.22T + 73T^{2} \)
79 \( 1 - 7.86T + 79T^{2} \)
83 \( 1 - 1.03T + 83T^{2} \)
89 \( 1 + 6.94T + 89T^{2} \)
97 \( 1 + 15.1T + 97T^{2} \)
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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

−8.431012115891253695023038122019, −7.86045991110388855330838761473, −7.05082362468888923147384513916, −6.07744940278611826599769921705, −5.68454614558465918116679158288, −5.06260500474489460378899734070, −3.49146698341575216169158487000, −2.79917806950955075894138892031, −1.81325517058211431969361441153, −0.945424236987175721142637123696, 0.945424236987175721142637123696, 1.81325517058211431969361441153, 2.79917806950955075894138892031, 3.49146698341575216169158487000, 5.06260500474489460378899734070, 5.68454614558465918116679158288, 6.07744940278611826599769921705, 7.05082362468888923147384513916, 7.86045991110388855330838761473, 8.431012115891253695023038122019

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