Properties

Label 2-257600-1.1-c1-0-15
Degree $2$
Conductor $257600$
Sign $1$
Analytic cond. $2056.94$
Root an. cond. $45.3535$
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 − 7-s − 2·9-s − 3·11-s + 3·13-s − 3·17-s − 2·19-s + 21-s + 23-s + 5·27-s − 3·29-s + 4·31-s + 3·33-s + 8·37-s − 3·39-s − 10·41-s − 4·43-s + 13·47-s + 49-s + 3·51-s + 2·57-s − 8·59-s + 10·61-s + 2·63-s + 6·67-s − 69-s − 6·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.832·13-s − 0.727·17-s − 0.458·19-s + 0.218·21-s + 0.208·23-s + 0.962·27-s − 0.557·29-s + 0.718·31-s + 0.522·33-s + 1.31·37-s − 0.480·39-s − 1.56·41-s − 0.609·43-s + 1.89·47-s + 1/7·49-s + 0.420·51-s + 0.264·57-s − 1.04·59-s + 1.28·61-s + 0.251·63-s + 0.733·67-s − 0.120·69-s − 0.702·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(257600\)    =    \(2^{6} \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(2056.94\)
Root analytic conductor: \(45.3535\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{257600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 257600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8128702539\)
\(L(\frac12)\) \(\approx\) \(0.8128702539\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good3 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−12.85306316183158, −12.36954748439057, −11.82016679490037, −11.40773611347919, −10.96044697225340, −10.67333849050320, −10.10783581083166, −9.703064777952804, −8.963747051822848, −8.647354778605560, −8.239748459599164, −7.692349537587410, −7.056308858404979, −6.557756539695975, −6.191599306422432, −5.627994002354908, −5.312903993414375, −4.650644023163458, −4.151890325900889, −3.554656520763113, −2.848802780649701, −2.545217070973881, −1.788826967984528, −0.9671137315540613, −0.2934469320971064, 0.2934469320971064, 0.9671137315540613, 1.788826967984528, 2.545217070973881, 2.848802780649701, 3.554656520763113, 4.151890325900889, 4.650644023163458, 5.312903993414375, 5.627994002354908, 6.191599306422432, 6.557756539695975, 7.056308858404979, 7.692349537587410, 8.239748459599164, 8.647354778605560, 8.963747051822848, 9.703064777952804, 10.10783581083166, 10.67333849050320, 10.96044697225340, 11.40773611347919, 11.82016679490037, 12.36954748439057, 12.85306316183158

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