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Properties

Label 2-305760-1.1-c1-0-12
Degree $2$
Conductor $305760$
Sign $1$
Analytic cond. $2441.50$
Root an. cond. $49.4115$
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  − 3-s + 5-s + 9-s + 3·11-s − 13-s − 15-s − 6·17-s − 8·19-s + 8·23-s + 25-s − 27-s − 8·29-s + 2·31-s − 3·33-s + 7·37-s + 39-s − 6·41-s + 12·43-s + 45-s − 6·47-s + 6·51-s − 9·53-s + 3·55-s + 8·57-s − 11·59-s − 9·61-s − 65-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 − 1.45·17-s − 1.83·19-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.48·29-s + 0.359·31-s − 0.522·33-s + 1.15·37-s + 0.160·39-s − 0.937·41-s + 1.82·43-s + 0.149·45-s − 0.875·47-s + 0.840·51-s − 1.23·53-s + 0.404·55-s + 1.05·57-s − 1.43·59-s − 1.15·61-s − 0.124·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 305760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 305760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(305760\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(2441.50\)
Root analytic conductor: \(49.4115\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 305760,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.172344848\)
\(L(\frac12)\) \(\approx\) \(1.172344848\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 9 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
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

−12.68601443549657, −12.34884203065264, −11.59146405684101, −11.25236624467795, −10.82129530835718, −10.66996023246768, −9.881698453704644, −9.319819653369460, −9.124584180386307, −8.687031030591227, −8.066536438036069, −7.360675886466097, −7.043507713111192, −6.366417341350925, −6.246944778864329, −5.766715091489663, −4.879825806180712, −4.613310211134504, −4.260308806833814, −3.520725340849960, −2.904366948010491, −2.176107661333650, −1.815063367912949, −1.120414705511643, −0.3088881609291710, 0.3088881609291710, 1.120414705511643, 1.815063367912949, 2.176107661333650, 2.904366948010491, 3.520725340849960, 4.260308806833814, 4.613310211134504, 4.879825806180712, 5.766715091489663, 6.246944778864329, 6.366417341350925, 7.043507713111192, 7.360675886466097, 8.066536438036069, 8.687031030591227, 9.124584180386307, 9.319819653369460, 9.881698453704644, 10.66996023246768, 10.82129530835718, 11.25236624467795, 11.59146405684101, 12.34884203065264, 12.68601443549657

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