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Properties

Label 2-369600-1.1-c1-0-571
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s + 9-s − 11-s − 6·13-s − 2·17-s − 21-s − 6·23-s − 27-s − 6·29-s − 2·31-s + 33-s + 10·37-s + 6·39-s − 8·41-s − 8·43-s − 4·47-s + 49-s + 2·51-s + 6·53-s + 6·59-s + 8·61-s + 63-s + 14·67-s + 6·69-s − 8·71-s + 2·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s − 0.485·17-s − 0.218·21-s − 1.25·23-s − 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.174·33-s + 1.64·37-s + 0.960·39-s − 1.24·41-s − 1.21·43-s − 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s + 0.781·59-s + 1.02·61-s + 0.125·63-s + 1.71·67-s + 0.722·69-s − 0.949·71-s + 0.234·73-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−12.95839092806722, −12.47935487820103, −11.91019738628927, −11.62814154759331, −11.27994914206180, −10.74722313850408, −10.07190340366055, −9.913499417077265, −9.545725890110634, −8.868022272144671, −8.194306008034007, −8.036026217217379, −7.318880484413350, −7.034089426535395, −6.553747990889985, −5.901123592996439, −5.394953625931790, −5.124735521641788, −4.525494886089497, −4.086870555377488, −3.547962864097093, −2.730556363939031, −2.224918117301398, −1.842721899481122, −1.039953754071340, 0, 0, 1.039953754071340, 1.842721899481122, 2.224918117301398, 2.730556363939031, 3.547962864097093, 4.086870555377488, 4.525494886089497, 5.124735521641788, 5.394953625931790, 5.901123592996439, 6.553747990889985, 7.034089426535395, 7.318880484413350, 8.036026217217379, 8.194306008034007, 8.868022272144671, 9.545725890110634, 9.913499417077265, 10.07190340366055, 10.74722313850408, 11.27994914206180, 11.62814154759331, 11.91019738628927, 12.47935487820103, 12.95839092806722

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