Properties

Label 2-50400-1.1-c1-0-95
Degree $2$
Conductor $50400$
Sign $-1$
Analytic cond. $402.446$
Root an. cond. $20.0610$
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  + 7-s + 4·13-s − 2·17-s − 2·19-s + 2·23-s − 4·29-s + 10·31-s − 6·37-s + 6·41-s + 4·43-s − 12·47-s + 49-s − 6·53-s − 12·59-s + 6·61-s − 4·67-s + 4·71-s + 8·73-s + 4·83-s − 18·89-s + 4·91-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.377·7-s + 1.10·13-s − 0.485·17-s − 0.458·19-s + 0.417·23-s − 0.742·29-s + 1.79·31-s − 0.986·37-s + 0.937·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s − 1.56·59-s + 0.768·61-s − 0.488·67-s + 0.474·71-s + 0.936·73-s + 0.439·83-s − 1.90·89-s + 0.419·91-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50400\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(402.446\)
Root analytic conductor: \(20.0610\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{50400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 50400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−14.73671577488822, −14.24859563281334, −13.68547438128345, −13.33246011874291, −12.72343542994802, −12.27798494159230, −11.54807157581950, −11.16736784254479, −10.74589010772319, −10.20806574830365, −9.455175796502027, −9.079094549679967, −8.363798399534579, −8.098059237492951, −7.441155151108598, −6.573464748174918, −6.421402680466698, −5.661749784479059, −5.005880346747745, −4.439950423987005, −3.861819857906175, −3.168459968874200, −2.488441483046044, −1.668019719644719, −1.069742258429793, 0, 1.069742258429793, 1.668019719644719, 2.488441483046044, 3.168459968874200, 3.861819857906175, 4.439950423987005, 5.005880346747745, 5.661749784479059, 6.421402680466698, 6.573464748174918, 7.441155151108598, 8.098059237492951, 8.363798399534579, 9.079094549679967, 9.455175796502027, 10.20806574830365, 10.74589010772319, 11.16736784254479, 11.54807157581950, 12.27798494159230, 12.72343542994802, 13.33246011874291, 13.68547438128345, 14.24859563281334, 14.73671577488822

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