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Properties

Label 2-76230-1.1-c1-0-81
Degree $2$
Conductor $76230$
Sign $-1$
Analytic cond. $608.699$
Root an. cond. $24.6718$
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 − 2·13-s − 14-s + 16-s − 2·19-s − 20-s + 25-s − 2·26-s − 28-s − 6·29-s + 8·31-s + 32-s + 35-s − 4·37-s − 2·38-s − 40-s − 6·41-s − 2·43-s − 6·47-s + 49-s + 50-s − 2·52-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 − 0.458·19-s − 0.223·20-s + 1/5·25-s − 0.392·26-s − 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.169·35-s − 0.657·37-s − 0.324·38-s − 0.158·40-s − 0.937·41-s − 0.304·43-s − 0.875·47-s + 1/7·49-s + 0.141·50-s − 0.277·52-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: $-1$
Analytic conductor: \(608.699\)
Root analytic conductor: \(24.6718\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
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 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.31327915240263, −13.69226199210069, −13.33944460031006, −12.84011889998387, −12.18891500652642, −12.05321840498191, −11.33081843852137, −10.99489475593170, −10.21445918595853, −9.951246177964758, −9.317715983777708, −8.510434451061889, −8.284180585750057, −7.423714591962800, −7.140420007529811, −6.461171028317357, −6.082791462547227, −5.267293922220844, −4.884774392891584, −4.281778788552810, −3.584982617640445, −3.257650285821476, −2.398072839087836, −1.919491448348124, −0.8915038587991809, 0, 0.8915038587991809, 1.919491448348124, 2.398072839087836, 3.257650285821476, 3.584982617640445, 4.281778788552810, 4.884774392891584, 5.267293922220844, 6.082791462547227, 6.461171028317357, 7.140420007529811, 7.423714591962800, 8.284180585750057, 8.510434451061889, 9.317715983777708, 9.951246177964758, 10.21445918595853, 10.99489475593170, 11.33081843852137, 12.05321840498191, 12.18891500652642, 12.84011889998387, 13.33944460031006, 13.69226199210069, 14.31327915240263

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