Properties

Label 2-185130-1.1-c1-0-111
Degree $2$
Conductor $185130$
Sign $-1$
Analytic cond. $1478.27$
Root an. cond. $38.4482$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s − 13-s + 14-s + 16-s − 17-s + 7·19-s + 20-s − 3·23-s + 25-s + 26-s − 28-s + 8·29-s − 5·31-s − 32-s + 34-s − 35-s + 5·37-s − 7·38-s − 40-s + 10·41-s − 2·43-s + 3·46-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 1.60·19-s + 0.223·20-s − 0.625·23-s + 1/5·25-s + 0.196·26-s − 0.188·28-s + 1.48·29-s − 0.898·31-s − 0.176·32-s + 0.171·34-s − 0.169·35-s + 0.821·37-s − 1.13·38-s − 0.158·40-s + 1.56·41-s − 0.304·43-s + 0.442·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185130\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(1478.27\)
Root analytic conductor: \(38.4482\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 185130,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17 \( 1 + T \)
good7 \( 1 + T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 9 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−13.29218076294302, −12.93863088998048, −12.26041941963679, −12.01563784910667, −11.39926320530833, −10.99288973504852, −10.42260548533930, −9.956503012525374, −9.495076662843770, −9.363712042458931, −8.569769422755587, −8.220802052507465, −7.538309919052986, −7.250446523709562, −6.637802007942487, −6.150162816333632, −5.661543292353209, −5.148257901919941, −4.519730376376319, −3.828919557491445, −3.230955811460451, −2.601247396609877, −2.242690150179831, −1.311638607372081, −0.8912923620656970, 0, 0.8912923620656970, 1.311638607372081, 2.242690150179831, 2.601247396609877, 3.230955811460451, 3.828919557491445, 4.519730376376319, 5.148257901919941, 5.661543292353209, 6.150162816333632, 6.637802007942487, 7.250446523709562, 7.538309919052986, 8.220802052507465, 8.569769422755587, 9.363712042458931, 9.495076662843770, 9.956503012525374, 10.42260548533930, 10.99288973504852, 11.39926320530833, 12.01563784910667, 12.26041941963679, 12.93863088998048, 13.29218076294302

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