Properties

Label 2-2070-1.1-c3-0-28
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 − 18.5·7-s + 8·8-s + 10·10-s − 47.9·11-s + 42.3·13-s − 37.0·14-s + 16·16-s − 1.70·17-s + 21.4·19-s + 20·20-s − 95.8·22-s + 23·23-s + 25·25-s + 84.7·26-s − 74.0·28-s − 57.6·29-s + 295.·31-s + 32·32-s − 3.41·34-s − 92.5·35-s − 7.85·37-s + 42.8·38-s + 40·40-s − 465.·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.999·7-s + 0.353·8-s + 0.316·10-s − 1.31·11-s + 0.903·13-s − 0.706·14-s + 0.250·16-s − 0.0243·17-s + 0.258·19-s + 0.223·20-s − 0.928·22-s + 0.208·23-s + 0.200·25-s + 0.639·26-s − 0.499·28-s − 0.369·29-s + 1.71·31-s + 0.176·32-s − 0.0172·34-s − 0.446·35-s − 0.0348·37-s + 0.182·38-s + 0.158·40-s − 1.77·41-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

\(L(2)\) \(\approx\) \(3.177991785\)
\(L(\frac12)\) \(\approx\) \(3.177991785\)
\(L(\frac{5}{2})\) 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 \( 1 + 18.5T + 343T^{2} \)
11 \( 1 + 47.9T + 1.33e3T^{2} \)
13 \( 1 - 42.3T + 2.19e3T^{2} \)
17 \( 1 + 1.70T + 4.91e3T^{2} \)
19 \( 1 - 21.4T + 6.85e3T^{2} \)
29 \( 1 + 57.6T + 2.43e4T^{2} \)
31 \( 1 - 295.T + 2.97e4T^{2} \)
37 \( 1 + 7.85T + 5.06e4T^{2} \)
41 \( 1 + 465.T + 6.89e4T^{2} \)
43 \( 1 - 182.T + 7.95e4T^{2} \)
47 \( 1 + 449.T + 1.03e5T^{2} \)
53 \( 1 - 368.T + 1.48e5T^{2} \)
59 \( 1 - 377.T + 2.05e5T^{2} \)
61 \( 1 - 849.T + 2.26e5T^{2} \)
67 \( 1 - 92.3T + 3.00e5T^{2} \)
71 \( 1 - 626.T + 3.57e5T^{2} \)
73 \( 1 - 439.T + 3.89e5T^{2} \)
79 \( 1 - 641.T + 4.93e5T^{2} \)
83 \( 1 - 609.T + 5.71e5T^{2} \)
89 \( 1 + 1.12e3T + 7.04e5T^{2} \)
97 \( 1 + 1.42e3T + 9.12e5T^{2} \)
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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

−8.651727894030545101978550966513, −7.996365657287612146796835646962, −6.90062763856064692564063746151, −6.37269538949155017563912160295, −5.53297577015500989333153686960, −4.88828795539854754750714780675, −3.68770339208868477079726375899, −3.00382639881459934394182280223, −2.11124055181872467287244434321, −0.71337618386922101725451085150, 0.71337618386922101725451085150, 2.11124055181872467287244434321, 3.00382639881459934394182280223, 3.68770339208868477079726375899, 4.88828795539854754750714780675, 5.53297577015500989333153686960, 6.37269538949155017563912160295, 6.90062763856064692564063746151, 7.996365657287612146796835646962, 8.651727894030545101978550966513

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