Properties

Label 2-53312-1.1-c1-0-46
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 4·5-s + 6·9-s + 11-s − 3·13-s + 12·15-s − 17-s − 2·19-s − 4·23-s + 11·25-s − 9·27-s − 8·31-s − 3·33-s − 8·37-s + 9·39-s + 10·43-s − 24·45-s + 10·47-s + 3·51-s − 3·53-s − 4·55-s + 6·57-s − 14·59-s + 8·61-s + 12·65-s − 10·67-s + 12·69-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.78·5-s + 2·9-s + 0.301·11-s − 0.832·13-s + 3.09·15-s − 0.242·17-s − 0.458·19-s − 0.834·23-s + 11/5·25-s − 1.73·27-s − 1.43·31-s − 0.522·33-s − 1.31·37-s + 1.44·39-s + 1.52·43-s − 3.57·45-s + 1.45·47-s + 0.420·51-s − 0.412·53-s − 0.539·55-s + 0.794·57-s − 1.82·59-s + 1.02·61-s + 1.48·65-s − 1.22·67-s + 1.44·69-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: \(2\)
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 \( 1 + p T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
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

−15.04719678659471, −14.64374705564196, −14.01820906763743, −13.13686155029263, −12.55034056986784, −12.24859079090191, −11.93060369543426, −11.48522930884121, −10.84653877832646, −10.66388011756630, −10.09594851730406, −9.149943814165438, −8.835617953744196, −7.867444827724194, −7.479908194552883, −7.124845272383791, −6.493833179210973, −5.902190192357147, −5.270622026328756, −4.727652512600304, −4.113517766131752, −3.894744503527745, −2.934266657571424, −1.911421389489433, −1.006067830773687, 0, 0, 1.006067830773687, 1.911421389489433, 2.934266657571424, 3.894744503527745, 4.113517766131752, 4.727652512600304, 5.270622026328756, 5.902190192357147, 6.493833179210973, 7.124845272383791, 7.479908194552883, 7.867444827724194, 8.835617953744196, 9.149943814165438, 10.09594851730406, 10.66388011756630, 10.84653877832646, 11.48522930884121, 11.93060369543426, 12.24859079090191, 12.55034056986784, 13.13686155029263, 14.01820906763743, 14.64374705564196, 15.04719678659471

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