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Properties

Label 2-76440-1.1-c1-0-54
Degree $2$
Conductor $76440$
Sign $-1$
Analytic cond. $610.376$
Root an. cond. $24.7057$
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  − 3-s + 5-s + 9-s + 3·11-s − 13-s − 15-s + 17-s + 19-s + 25-s − 27-s − 5·29-s − 4·31-s − 3·33-s + 2·37-s + 39-s − 2·41-s − 10·43-s + 45-s + 7·47-s − 51-s − 3·53-s + 3·55-s − 57-s − 5·59-s − 61-s − 65-s + 9·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.904·11-s − 0.277·13-s − 0.258·15-s + 0.242·17-s + 0.229·19-s + 1/5·25-s − 0.192·27-s − 0.928·29-s − 0.718·31-s − 0.522·33-s + 0.328·37-s + 0.160·39-s − 0.312·41-s − 1.52·43-s + 0.149·45-s + 1.02·47-s − 0.140·51-s − 0.412·53-s + 0.404·55-s − 0.132·57-s − 0.650·59-s − 0.128·61-s − 0.124·65-s + 1.09·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76440\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(610.376\)
Root analytic conductor: \(24.7057\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 76440,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 \)
13 \( 1 + T \)
good11 \( 1 - 3 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 11 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−14.32578850512401, −13.78873135907083, −13.26663566022549, −12.74436712201486, −12.31251252062169, −11.76278364498357, −11.31988641928392, −10.89766349392832, −10.21694637798967, −9.788824580858115, −9.333363287449713, −8.843851285027477, −8.185386949033950, −7.546609132221520, −6.976693738061325, −6.617129995966367, −5.870327954027422, −5.577963944553792, −4.904060244650596, −4.345297732527706, −3.652907929174430, −3.146126099366834, −2.180220830326035, −1.649236755999733, −0.9380610494713711, 0, 0.9380610494713711, 1.649236755999733, 2.180220830326035, 3.146126099366834, 3.652907929174430, 4.345297732527706, 4.904060244650596, 5.577963944553792, 5.870327954027422, 6.617129995966367, 6.976693738061325, 7.546609132221520, 8.185386949033950, 8.843851285027477, 9.333363287449713, 9.788824580858115, 10.21694637798967, 10.89766349392832, 11.31988641928392, 11.76278364498357, 12.31251252062169, 12.74436712201486, 13.26663566022549, 13.78873135907083, 14.32578850512401

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