Properties

Label 2-2070-1.1-c1-0-4
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 + 1.61·7-s − 8-s + 10-s − 3.85·11-s + 4.09·13-s − 1.61·14-s + 16-s + 5.09·17-s − 4.85·19-s − 20-s + 3.85·22-s − 23-s + 25-s − 4.09·26-s + 1.61·28-s + 4.76·29-s − 2.09·31-s − 32-s − 5.09·34-s − 1.61·35-s − 2.47·37-s + 4.85·38-s + 40-s + 12.3·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.611·7-s − 0.353·8-s + 0.316·10-s − 1.16·11-s + 1.13·13-s − 0.432·14-s + 0.250·16-s + 1.23·17-s − 1.11·19-s − 0.223·20-s + 0.821·22-s − 0.208·23-s + 0.200·25-s − 0.802·26-s + 0.305·28-s + 0.884·29-s − 0.375·31-s − 0.176·32-s − 0.872·34-s − 0.273·35-s − 0.406·37-s + 0.787·38-s + 0.158·40-s + 1.92·41-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

\(L(1)\) \(\approx\) \(1.170604794\)
\(L(\frac12)\) \(\approx\) \(1.170604794\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.61T + 7T^{2} \)
11 \( 1 + 3.85T + 11T^{2} \)
13 \( 1 - 4.09T + 13T^{2} \)
17 \( 1 - 5.09T + 17T^{2} \)
19 \( 1 + 4.85T + 19T^{2} \)
29 \( 1 - 4.76T + 29T^{2} \)
31 \( 1 + 2.09T + 31T^{2} \)
37 \( 1 + 2.47T + 37T^{2} \)
41 \( 1 - 12.3T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 9.70T + 47T^{2} \)
53 \( 1 - 8.47T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 - 6.32T + 61T^{2} \)
67 \( 1 - 5.52T + 67T^{2} \)
71 \( 1 + 7.09T + 71T^{2} \)
73 \( 1 + 1.23T + 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 - 1.52T + 89T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
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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

−8.915919089267394368139024097000, −8.193832656684335668977110465955, −7.919274553111697184622416661461, −6.95851054108973961374312161214, −6.00760005514998738072551903495, −5.22464210615257006858018019389, −4.15340517730267265336944226327, −3.15578636193988013655177999170, −2.05326656139358273818644960548, −0.803817388064460988594638353418, 0.803817388064460988594638353418, 2.05326656139358273818644960548, 3.15578636193988013655177999170, 4.15340517730267265336944226327, 5.22464210615257006858018019389, 6.00760005514998738072551903495, 6.95851054108973961374312161214, 7.919274553111697184622416661461, 8.193832656684335668977110465955, 8.915919089267394368139024097000

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