Properties

Label 2-369600-1.1-c1-0-140
Degree $2$
Conductor $369600$
Sign $1$
Analytic cond. $2951.27$
Root an. cond. $54.3256$
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 + 9-s − 11-s − 4·13-s + 4·19-s − 21-s − 6·23-s + 27-s − 6·29-s − 4·31-s − 33-s + 8·37-s − 4·39-s + 6·41-s + 8·43-s + 49-s + 6·53-s + 4·57-s + 10·61-s − 63-s + 2·67-s − 6·69-s + 12·71-s + 10·73-s + 77-s − 4·79-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 0.917·19-s − 0.218·21-s − 1.25·23-s + 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.174·33-s + 1.31·37-s − 0.640·39-s + 0.937·41-s + 1.21·43-s + 1/7·49-s + 0.824·53-s + 0.529·57-s + 1.28·61-s − 0.125·63-s + 0.244·67-s − 0.722·69-s + 1.42·71-s + 1.17·73-s + 0.113·77-s − 0.450·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(2951.27\)
Root analytic conductor: \(54.3256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{369600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.768348123\)
\(L(\frac12)\) \(\approx\) \(2.768348123\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
good13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.41091232385757, −12.31934934877633, −11.48517543210202, −11.26827751111941, −10.63372421852718, −10.08425312209814, −9.734471211978351, −9.318908364847313, −9.103856375664840, −8.241126347843809, −7.899142786341728, −7.507257977832738, −7.132695420528131, −6.569885245017283, −5.888399765273239, −5.539128390923643, −5.055745893009420, −4.337614499134897, −3.943637637251373, −3.454738926591384, −2.821582602449259, −2.197108979856299, −2.088399211014417, −0.9899552139912670, −0.4588160761623261, 0.4588160761623261, 0.9899552139912670, 2.088399211014417, 2.197108979856299, 2.821582602449259, 3.454738926591384, 3.943637637251373, 4.337614499134897, 5.055745893009420, 5.539128390923643, 5.888399765273239, 6.569885245017283, 7.132695420528131, 7.507257977832738, 7.899142786341728, 8.241126347843809, 9.103856375664840, 9.318908364847313, 9.734471211978351, 10.08425312209814, 10.63372421852718, 11.26827751111941, 11.48517543210202, 12.31934934877633, 12.41091232385757

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