Properties

Label 2-4030-1.1-c1-0-63
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 + 1.68·3-s + 4-s − 5-s + 1.68·6-s + 2.51·7-s + 8-s − 0.147·9-s − 10-s + 4.10·11-s + 1.68·12-s + 13-s + 2.51·14-s − 1.68·15-s + 16-s − 1.63·17-s − 0.147·18-s − 0.771·19-s − 20-s + 4.25·21-s + 4.10·22-s + 7.81·23-s + 1.68·24-s + 25-s + 26-s − 5.31·27-s + 2.51·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.975·3-s + 0.5·4-s − 0.447·5-s + 0.689·6-s + 0.951·7-s + 0.353·8-s − 0.0491·9-s − 0.316·10-s + 1.23·11-s + 0.487·12-s + 0.277·13-s + 0.673·14-s − 0.436·15-s + 0.250·16-s − 0.397·17-s − 0.0347·18-s − 0.176·19-s − 0.223·20-s + 0.928·21-s + 0.875·22-s + 1.62·23-s + 0.344·24-s + 0.200·25-s + 0.196·26-s − 1.02·27-s + 0.475·28-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

\(L(1)\) \(\approx\) \(4.830064217\)
\(L(\frac12)\) \(\approx\) \(4.830064217\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.68T + 3T^{2} \)
7 \( 1 - 2.51T + 7T^{2} \)
11 \( 1 - 4.10T + 11T^{2} \)
17 \( 1 + 1.63T + 17T^{2} \)
19 \( 1 + 0.771T + 19T^{2} \)
23 \( 1 - 7.81T + 23T^{2} \)
29 \( 1 - 2.36T + 29T^{2} \)
37 \( 1 + 5.66T + 37T^{2} \)
41 \( 1 - 1.56T + 41T^{2} \)
43 \( 1 - 6.86T + 43T^{2} \)
47 \( 1 + 12.5T + 47T^{2} \)
53 \( 1 - 5.08T + 53T^{2} \)
59 \( 1 - 6.23T + 59T^{2} \)
61 \( 1 + 3.23T + 61T^{2} \)
67 \( 1 - 8.51T + 67T^{2} \)
71 \( 1 - 2.05T + 71T^{2} \)
73 \( 1 + 3.66T + 73T^{2} \)
79 \( 1 - 3.03T + 79T^{2} \)
83 \( 1 + 8.01T + 83T^{2} \)
89 \( 1 - 5.78T + 89T^{2} \)
97 \( 1 - 7.69T + 97T^{2} \)
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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

−8.565088236761156093991495981754, −7.72461099954071528937140885649, −6.99960607671079582230249217393, −6.31456893043472021406421596377, −5.24855753238923014279366422494, −4.55112872821513872620231866939, −3.78420527020811303511199568672, −3.12618930187154019550215016805, −2.16090180720807710949361133117, −1.19280338956479130910494335836, 1.19280338956479130910494335836, 2.16090180720807710949361133117, 3.12618930187154019550215016805, 3.78420527020811303511199568672, 4.55112872821513872620231866939, 5.24855753238923014279366422494, 6.31456893043472021406421596377, 6.99960607671079582230249217393, 7.72461099954071528937140885649, 8.565088236761156093991495981754

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