Properties

Label 2-141570-1.1-c1-0-22
Degree $2$
Conductor $141570$
Sign $1$
Analytic cond. $1130.44$
Root an. cond. $33.6220$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 3·7-s + 8-s − 10-s − 13-s − 3·14-s + 16-s − 3·17-s + 4·19-s − 20-s + 8·23-s + 25-s − 26-s − 3·28-s + 9·29-s − 4·31-s + 32-s − 3·34-s + 3·35-s − 2·37-s + 4·38-s − 40-s − 10·41-s + 11·43-s + 8·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.13·7-s + 0.353·8-s − 0.316·10-s − 0.277·13-s − 0.801·14-s + 1/4·16-s − 0.727·17-s + 0.917·19-s − 0.223·20-s + 1.66·23-s + 1/5·25-s − 0.196·26-s − 0.566·28-s + 1.67·29-s − 0.718·31-s + 0.176·32-s − 0.514·34-s + 0.507·35-s − 0.328·37-s + 0.648·38-s − 0.158·40-s − 1.56·41-s + 1.67·43-s + 1.17·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(141570\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(1130.44\)
Root analytic conductor: \(33.6220\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{141570} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 141570,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.515718498\)
\(L(\frac12)\) \(\approx\) \(2.515718498\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.39305749174529, −12.90562930670615, −12.52342623619220, −12.01240197431221, −11.69370859883360, −10.95631369477344, −10.67353360056494, −10.13715407299369, −9.463267126972963, −9.112980643829802, −8.625506395887154, −7.877345133434991, −7.314238002573500, −7.019023605284911, −6.343363196871831, −6.148071016327690, −5.270036812687205, −4.767891805091567, −4.491400363408889, −3.537373257626389, −3.186615310716629, −2.883528524678222, −2.040342229378243, −1.204420303710941, −0.4357960398624866, 0.4357960398624866, 1.204420303710941, 2.040342229378243, 2.883528524678222, 3.186615310716629, 3.537373257626389, 4.491400363408889, 4.767891805091567, 5.270036812687205, 6.148071016327690, 6.343363196871831, 7.019023605284911, 7.314238002573500, 7.877345133434991, 8.625506395887154, 9.112980643829802, 9.463267126972963, 10.13715407299369, 10.67353360056494, 10.95631369477344, 11.69370859883360, 12.01240197431221, 12.52342623619220, 12.90562930670615, 13.39305749174529

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