Properties

Label 2-53312-1.1-c1-0-53
Degree $2$
Conductor $53312$
Sign $-1$
Analytic cond. $425.698$
Root an. cond. $20.6324$
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 + 4·5-s − 2·9-s + 3·11-s + 5·13-s − 4·15-s − 17-s − 6·19-s − 4·23-s + 11·25-s + 5·27-s − 8·29-s − 3·33-s + 8·37-s − 5·39-s − 8·41-s − 10·43-s − 8·45-s − 2·47-s + 51-s − 3·53-s + 12·55-s + 6·57-s − 2·59-s + 8·61-s + 20·65-s − 14·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.78·5-s − 2/3·9-s + 0.904·11-s + 1.38·13-s − 1.03·15-s − 0.242·17-s − 1.37·19-s − 0.834·23-s + 11/5·25-s + 0.962·27-s − 1.48·29-s − 0.522·33-s + 1.31·37-s − 0.800·39-s − 1.24·41-s − 1.52·43-s − 1.19·45-s − 0.291·47-s + 0.140·51-s − 0.412·53-s + 1.61·55-s + 0.794·57-s − 0.260·59-s + 1.02·61-s + 2.48·65-s − 1.71·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(53312\)    =    \(2^{6} \cdot 7^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(425.698\)
Root analytic conductor: \(20.6324\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 53312,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
17 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−14.61782797106948, −14.21170920252842, −13.53230522644998, −13.25746473755674, −12.88563417481234, −12.09020429319721, −11.52334866161896, −11.12138605610018, −10.59005766063634, −10.11997369239722, −9.542425440355935, −8.995929421060638, −8.608306720116124, −8.111896257529636, −7.051328401186364, −6.396858931473881, −6.183690507192599, −5.871454144885252, −5.181773837877558, −4.554360123512507, −3.765321086840835, −3.154852083660407, −2.175685724518195, −1.804442999154711, −1.104045002630146, 0, 1.104045002630146, 1.804442999154711, 2.175685724518195, 3.154852083660407, 3.765321086840835, 4.554360123512507, 5.181773837877558, 5.871454144885252, 6.183690507192599, 6.396858931473881, 7.051328401186364, 8.111896257529636, 8.608306720116124, 8.995929421060638, 9.542425440355935, 10.11997369239722, 10.59005766063634, 11.12138605610018, 11.52334866161896, 12.09020429319721, 12.88563417481234, 13.25746473755674, 13.53230522644998, 14.21170920252842, 14.61782797106948

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