Properties

Label 2-4030-1.1-c1-0-39
Degree $2$
Conductor $4030$
Sign $1$
Analytic cond. $32.1797$
Root an. cond. $5.67271$
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 + 0.161·3-s + 4-s − 5-s + 0.161·6-s + 0.164·7-s + 8-s − 2.97·9-s − 10-s + 2.35·11-s + 0.161·12-s + 13-s + 0.164·14-s − 0.161·15-s + 16-s − 3.98·17-s − 2.97·18-s + 7.89·19-s − 20-s + 0.0264·21-s + 2.35·22-s − 3.21·23-s + 0.161·24-s + 25-s + 26-s − 0.964·27-s + 0.164·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.0931·3-s + 0.5·4-s − 0.447·5-s + 0.0658·6-s + 0.0620·7-s + 0.353·8-s − 0.991·9-s − 0.316·10-s + 0.708·11-s + 0.0465·12-s + 0.277·13-s + 0.0438·14-s − 0.0416·15-s + 0.250·16-s − 0.965·17-s − 0.700·18-s + 1.81·19-s − 0.223·20-s + 0.00577·21-s + 0.501·22-s − 0.670·23-s + 0.0329·24-s + 0.200·25-s + 0.196·26-s − 0.185·27-s + 0.0310·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4030\)    =    \(2 \cdot 5 \cdot 13 \cdot 31\)
Sign: $1$
Analytic conductor: \(32.1797\)
Root analytic conductor: \(5.67271\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4030,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.864056097\)
\(L(\frac12)\) \(\approx\) \(2.864056097\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 - T \)
31 \( 1 - T \)
good3 \( 1 - 0.161T + 3T^{2} \)
7 \( 1 - 0.164T + 7T^{2} \)
11 \( 1 - 2.35T + 11T^{2} \)
17 \( 1 + 3.98T + 17T^{2} \)
19 \( 1 - 7.89T + 19T^{2} \)
23 \( 1 + 3.21T + 23T^{2} \)
29 \( 1 - 10.4T + 29T^{2} \)
37 \( 1 - 0.957T + 37T^{2} \)
41 \( 1 - 3.80T + 41T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 - 3.52T + 47T^{2} \)
53 \( 1 + 5.93T + 53T^{2} \)
59 \( 1 - 4.00T + 59T^{2} \)
61 \( 1 - 6.23T + 61T^{2} \)
67 \( 1 + 1.88T + 67T^{2} \)
71 \( 1 - 8.81T + 71T^{2} \)
73 \( 1 - 15.5T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 - 13.9T + 89T^{2} \)
97 \( 1 - 9.61T + 97T^{2} \)
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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

−8.251430289727680649520081299775, −7.82689981189740468994629117181, −6.64018880466484488672503892140, −6.40433113948798909888750304049, −5.29683663370449361711474933847, −4.75086824867708307750725464268, −3.73248573101481237449953252672, −3.16777570505801057732504276925, −2.20528882255948944581608683650, −0.876445763163244354104558612460, 0.876445763163244354104558612460, 2.20528882255948944581608683650, 3.16777570505801057732504276925, 3.73248573101481237449953252672, 4.75086824867708307750725464268, 5.29683663370449361711474933847, 6.40433113948798909888750304049, 6.64018880466484488672503892140, 7.82689981189740468994629117181, 8.251430289727680649520081299775

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