Properties

Label 2-92400-1.1-c1-0-3
Degree $2$
Conductor $92400$
Sign $1$
Analytic cond. $737.817$
Root an. cond. $27.1628$
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 − 6·13-s − 2·17-s + 21-s − 27-s − 6·29-s + 8·31-s + 33-s − 10·37-s + 6·39-s − 2·41-s − 8·43-s + 4·47-s + 49-s + 2·51-s + 6·53-s + 12·59-s + 10·61-s − 63-s + 8·67-s − 4·71-s − 10·73-s + 77-s − 8·79-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 − 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.174·33-s − 1.64·37-s + 0.960·39-s − 0.312·41-s − 1.21·43-s + 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s + 1.56·59-s + 1.28·61-s − 0.125·63-s + 0.977·67-s − 0.474·71-s − 1.17·73-s + 0.113·77-s − 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(737.817\)
Root analytic conductor: \(27.1628\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4108369038\)
\(L(\frac12)\) \(\approx\) \(0.4108369038\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.65877238352823, −13.32974706658765, −12.85660682871762, −12.24986145129729, −11.89370194186209, −11.55100014314962, −10.81618477052145, −10.32486910792088, −9.864712734847206, −9.640602880815785, −8.717575159497723, −8.471228138509505, −7.602703604010460, −7.191067485539631, −6.779080502969333, −6.251315186632023, −5.387375446413740, −5.225036801358448, −4.599126281354079, −3.930372241171712, −3.334812073104876, −2.457469185691058, −2.166393962430380, −1.168976587516042, −0.2208810302127918, 0.2208810302127918, 1.168976587516042, 2.166393962430380, 2.457469185691058, 3.334812073104876, 3.930372241171712, 4.599126281354079, 5.225036801358448, 5.387375446413740, 6.251315186632023, 6.779080502969333, 7.191067485539631, 7.602703604010460, 8.471228138509505, 8.717575159497723, 9.640602880815785, 9.864712734847206, 10.32486910792088, 10.81618477052145, 11.55100014314962, 11.89370194186209, 12.24986145129729, 12.85660682871762, 13.32974706658765, 13.65877238352823

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