Properties

Label 2-2070-1.1-c3-0-61
Degree $2$
Conductor $2070$
Sign $1$
Analytic cond. $122.133$
Root an. cond. $11.0514$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·4-s + 5·5-s + 23.5·7-s + 8·8-s + 10·10-s + 32.1·11-s + 40.0·13-s + 47.1·14-s + 16·16-s − 126.·17-s + 0.232·19-s + 20·20-s + 64.2·22-s + 23·23-s + 25·25-s + 80.1·26-s + 94.2·28-s + 137.·29-s + 112.·31-s + 32·32-s − 252.·34-s + 117.·35-s + 45.7·37-s + 0.465·38-s + 40·40-s + 135.·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.27·7-s + 0.353·8-s + 0.316·10-s + 0.880·11-s + 0.855·13-s + 0.899·14-s + 0.250·16-s − 1.79·17-s + 0.00281·19-s + 0.223·20-s + 0.622·22-s + 0.208·23-s + 0.200·25-s + 0.604·26-s + 0.636·28-s + 0.878·29-s + 0.653·31-s + 0.176·32-s − 1.27·34-s + 0.568·35-s + 0.203·37-s + 0.00198·38-s + 0.158·40-s + 0.515·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(122.133\)
Root analytic conductor: \(11.0514\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2070} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.449887161\)
\(L(\frac12)\) \(\approx\) \(5.449887161\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 \)
5 \( 1 - 5T \)
23 \( 1 - 23T \)
good7 \( 1 - 23.5T + 343T^{2} \)
11 \( 1 - 32.1T + 1.33e3T^{2} \)
13 \( 1 - 40.0T + 2.19e3T^{2} \)
17 \( 1 + 126.T + 4.91e3T^{2} \)
19 \( 1 - 0.232T + 6.85e3T^{2} \)
29 \( 1 - 137.T + 2.43e4T^{2} \)
31 \( 1 - 112.T + 2.97e4T^{2} \)
37 \( 1 - 45.7T + 5.06e4T^{2} \)
41 \( 1 - 135.T + 6.89e4T^{2} \)
43 \( 1 - 543.T + 7.95e4T^{2} \)
47 \( 1 + 26.4T + 1.03e5T^{2} \)
53 \( 1 + 43.6T + 1.48e5T^{2} \)
59 \( 1 + 202.T + 2.05e5T^{2} \)
61 \( 1 - 150.T + 2.26e5T^{2} \)
67 \( 1 + 420.T + 3.00e5T^{2} \)
71 \( 1 + 667.T + 3.57e5T^{2} \)
73 \( 1 - 602.T + 3.89e5T^{2} \)
79 \( 1 + 1.37e3T + 4.93e5T^{2} \)
83 \( 1 - 485.T + 5.71e5T^{2} \)
89 \( 1 - 1.12e3T + 7.04e5T^{2} \)
97 \( 1 + 1.48e3T + 9.12e5T^{2} \)
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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

−8.734639102398223464729398932997, −8.025310317113633411150144011704, −6.97444959812780281197566438896, −6.33333455761908934731423491798, −5.58064050892533669009889014414, −4.47438992838127563222157657593, −4.23867192627264129939269951892, −2.82994716033130175391670190752, −1.88918398208748364723503089125, −1.04067909133511062360647544350, 1.04067909133511062360647544350, 1.88918398208748364723503089125, 2.82994716033130175391670190752, 4.23867192627264129939269951892, 4.47438992838127563222157657593, 5.58064050892533669009889014414, 6.33333455761908934731423491798, 6.97444959812780281197566438896, 8.025310317113633411150144011704, 8.734639102398223464729398932997

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