Properties

Label 2-2070-1.1-c1-0-11
Degree $2$
Conductor $2070$
Sign $1$
Analytic cond. $16.5290$
Root an. cond. $4.06559$
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 − 5-s − 0.302·7-s + 8-s − 10-s + 5.30·11-s − 0.302·13-s − 0.302·14-s + 16-s + 3.90·17-s − 4.90·19-s − 20-s + 5.30·22-s + 23-s + 25-s − 0.302·26-s − 0.302·28-s − 4.60·29-s + 2.90·31-s + 32-s + 3.90·34-s + 0.302·35-s + 8·37-s − 4.90·38-s − 40-s + 9.90·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.114·7-s + 0.353·8-s − 0.316·10-s + 1.59·11-s − 0.0839·13-s − 0.0809·14-s + 0.250·16-s + 0.947·17-s − 1.12·19-s − 0.223·20-s + 1.13·22-s + 0.208·23-s + 0.200·25-s − 0.0593·26-s − 0.0572·28-s − 0.855·29-s + 0.522·31-s + 0.176·32-s + 0.670·34-s + 0.0511·35-s + 1.31·37-s − 0.796·38-s − 0.158·40-s + 1.54·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(16.5290\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2070} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.829811228\)
\(L(\frac12)\) \(\approx\) \(2.829811228\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
23 \( 1 - T \)
good7 \( 1 + 0.302T + 7T^{2} \)
11 \( 1 - 5.30T + 11T^{2} \)
13 \( 1 + 0.302T + 13T^{2} \)
17 \( 1 - 3.90T + 17T^{2} \)
19 \( 1 + 4.90T + 19T^{2} \)
29 \( 1 + 4.60T + 29T^{2} \)
31 \( 1 - 2.90T + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 - 9.90T + 41T^{2} \)
43 \( 1 - 5.21T + 43T^{2} \)
47 \( 1 + 4.60T + 47T^{2} \)
53 \( 1 + 3.21T + 53T^{2} \)
59 \( 1 - 10.6T + 59T^{2} \)
61 \( 1 + 6.51T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 - 15.8T + 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 - 3.21T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 2.69T + 97T^{2} \)
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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

−9.228192260236810906156805897813, −8.191381251989494102129154696399, −7.51337932851913533906279946267, −6.52754052796858718914512777567, −6.11311939324029171015771297191, −4.97698543825620063644638945037, −4.08996027210217114618458707711, −3.56612978338512225674629872144, −2.36505019626353441922533065140, −1.06854396603220431591782468674, 1.06854396603220431591782468674, 2.36505019626353441922533065140, 3.56612978338512225674629872144, 4.08996027210217114618458707711, 4.97698543825620063644638945037, 6.11311939324029171015771297191, 6.52754052796858718914512777567, 7.51337932851913533906279946267, 8.191381251989494102129154696399, 9.228192260236810906156805897813

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