Properties

Degree $2$
Conductor $76230$
Sign $-1$
Motivic weight $1$
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 − 2·13-s − 14-s + 16-s − 6·17-s + 4·19-s + 20-s + 25-s − 2·26-s − 28-s − 6·29-s − 4·31-s + 32-s − 6·34-s − 35-s + 2·37-s + 4·38-s + 40-s + 6·41-s − 8·43-s + 12·47-s + 49-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 − 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.223·20-s + 1/5·25-s − 0.392·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.169·35-s + 0.328·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76230\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{76230} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 76230,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.30447878015427, −13.66737023470917, −13.23227845788977, −13.05987569318826, −12.30318970518135, −11.96295567929800, −11.26806951097428, −10.92819361057670, −10.38463345318209, −9.703652758861543, −9.291767162888411, −8.914059856924961, −8.040967940482423, −7.550285712281634, −6.877228363475275, −6.670127126193705, −5.816997187389043, −5.501554860166996, −4.892004524537578, −4.259625998251777, −3.717926453149256, −3.055231009147415, −2.341344003847685, −1.981427845539001, −0.9929531024512531, 0, 0.9929531024512531, 1.981427845539001, 2.341344003847685, 3.055231009147415, 3.717926453149256, 4.259625998251777, 4.892004524537578, 5.501554860166996, 5.816997187389043, 6.670127126193705, 6.877228363475275, 7.550285712281634, 8.040967940482423, 8.914059856924961, 9.291767162888411, 9.703652758861543, 10.38463345318209, 10.92819361057670, 11.26806951097428, 11.96295567929800, 12.30318970518135, 13.05987569318826, 13.23227845788977, 13.66737023470917, 14.30447878015427

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