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Properties

Label 2-388080-1.1-c1-0-101
Degree $2$
Conductor $388080$
Sign $1$
Analytic cond. $3098.83$
Root an. cond. $55.6671$
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  + 5-s + 11-s + 2·13-s + 17-s − 7·19-s − 3·23-s + 25-s + 9·29-s − 8·31-s − 2·37-s − 6·41-s + 9·43-s + 6·47-s − 13·53-s + 55-s + 11·59-s + 13·61-s + 2·65-s + 4·67-s − 6·71-s − 6·73-s + 12·79-s − 83-s + 85-s − 5·89-s − 7·95-s − 15·97-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.301·11-s + 0.554·13-s + 0.242·17-s − 1.60·19-s − 0.625·23-s + 1/5·25-s + 1.67·29-s − 1.43·31-s − 0.328·37-s − 0.937·41-s + 1.37·43-s + 0.875·47-s − 1.78·53-s + 0.134·55-s + 1.43·59-s + 1.66·61-s + 0.248·65-s + 0.488·67-s − 0.712·71-s − 0.702·73-s + 1.35·79-s − 0.109·83-s + 0.108·85-s − 0.529·89-s − 0.718·95-s − 1.52·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(388080\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(3098.83\)
Root analytic conductor: \(55.6671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 388080,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.788317509\)
\(L(\frac12)\) \(\approx\) \(2.788317509\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
11 \( 1 - T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 13 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 + 15 T + p T^{2} \)
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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

−12.60308976962794, −12.07046950228697, −11.51513233460064, −11.07764809905314, −10.55155919761967, −10.35022258073269, −9.741838786390309, −9.319835485642540, −8.733945018126789, −8.436250477952600, −8.073832467502582, −7.325903163618266, −6.867412053021714, −6.463367184567001, −5.983219306735513, −5.580718522170521, −5.009951137204331, −4.351565100120417, −4.046147852375266, −3.433514806468772, −2.871123073860937, −2.108259695507636, −1.897078374153811, −1.072930041701754, −0.4547592662832419, 0.4547592662832419, 1.072930041701754, 1.897078374153811, 2.108259695507636, 2.871123073860937, 3.433514806468772, 4.046147852375266, 4.351565100120417, 5.009951137204331, 5.580718522170521, 5.983219306735513, 6.463367184567001, 6.867412053021714, 7.325903163618266, 8.073832467502582, 8.436250477952600, 8.733945018126789, 9.319835485642540, 9.741838786390309, 10.35022258073269, 10.55155919761967, 11.07764809905314, 11.51513233460064, 12.07046950228697, 12.60308976962794

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