Properties

Label 2-2070-1.1-c1-0-16
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 + 3.30·7-s + 8-s − 10-s + 1.69·11-s + 3.30·13-s + 3.30·14-s + 16-s − 6.90·17-s + 5.90·19-s − 20-s + 1.69·22-s + 23-s + 25-s + 3.30·26-s + 3.30·28-s + 2.60·29-s − 7.90·31-s + 32-s − 6.90·34-s − 3.30·35-s + 8·37-s + 5.90·38-s − 40-s − 0.908·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.24·7-s + 0.353·8-s − 0.316·10-s + 0.511·11-s + 0.916·13-s + 0.882·14-s + 0.250·16-s − 1.67·17-s + 1.35·19-s − 0.223·20-s + 0.361·22-s + 0.208·23-s + 0.200·25-s + 0.647·26-s + 0.624·28-s + 0.483·29-s − 1.42·31-s + 0.176·32-s − 1.18·34-s − 0.558·35-s + 1.31·37-s + 0.958·38-s − 0.158·40-s − 0.141·41-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

\(L(1)\) \(\approx\) \(3.195543055\)
\(L(\frac12)\) \(\approx\) \(3.195543055\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.30T + 7T^{2} \)
11 \( 1 - 1.69T + 11T^{2} \)
13 \( 1 - 3.30T + 13T^{2} \)
17 \( 1 + 6.90T + 17T^{2} \)
19 \( 1 - 5.90T + 19T^{2} \)
29 \( 1 - 2.60T + 29T^{2} \)
31 \( 1 + 7.90T + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + 0.908T + 41T^{2} \)
43 \( 1 + 9.21T + 43T^{2} \)
47 \( 1 - 2.60T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 - 3.39T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 16.3T + 71T^{2} \)
73 \( 1 + 5.81T + 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 + 11.2T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 6.30T + 97T^{2} \)
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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

−8.852819216778846306919084780360, −8.395717147071233166762378179916, −7.41609433511581774568989715472, −6.81682431454569900359642108095, −5.81021230889943320094525581531, −4.99000246623524088649143621828, −4.25771499872425734760953895160, −3.51116061532589262500833595437, −2.25339301567425683445484211469, −1.17981764933870209034850387785, 1.17981764933870209034850387785, 2.25339301567425683445484211469, 3.51116061532589262500833595437, 4.25771499872425734760953895160, 4.99000246623524088649143621828, 5.81021230889943320094525581531, 6.81682431454569900359642108095, 7.41609433511581774568989715472, 8.395717147071233166762378179916, 8.852819216778846306919084780360

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