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Properties

Label 2-283920-1.1-c1-0-35
Degree $2$
Conductor $283920$
Sign $1$
Analytic cond. $2267.11$
Root an. cond. $47.6142$
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 + 5-s − 7-s + 9-s − 4·11-s + 15-s − 6·17-s − 21-s + 8·23-s + 25-s + 27-s + 10·29-s − 8·31-s − 4·33-s − 35-s − 2·37-s + 2·41-s − 8·43-s + 45-s + 4·47-s + 49-s − 6·51-s + 10·53-s − 4·55-s + 4·59-s − 6·61-s − 63-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.258·15-s − 1.45·17-s − 0.218·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.85·29-s − 1.43·31-s − 0.696·33-s − 0.169·35-s − 0.328·37-s + 0.312·41-s − 1.21·43-s + 0.149·45-s + 0.583·47-s + 1/7·49-s − 0.840·51-s + 1.37·53-s − 0.539·55-s + 0.520·59-s − 0.768·61-s − 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(283920\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(2267.11\)
Root analytic conductor: \(47.6142\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 283920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.409567981\)
\(L(\frac12)\) \(\approx\) \(2.409567981\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−12.74673430205200, −12.62035376371012, −11.77251382875248, −11.33054992250521, −10.70500806944401, −10.46401546582762, −10.07204244849483, −9.372060472051871, −9.059850001737780, −8.482519853553075, −8.395251545980919, −7.366181860954229, −7.271761285327994, −6.658277973360080, −6.248593623100655, −5.487890515478721, −5.132374412263565, −4.598396871126390, −4.112484586468926, −3.269621619873820, −2.960071701109125, −2.393104012335151, −1.973131666579210, −1.146770309051049, −0.4069912163468042, 0.4069912163468042, 1.146770309051049, 1.973131666579210, 2.393104012335151, 2.960071701109125, 3.269621619873820, 4.112484586468926, 4.598396871126390, 5.132374412263565, 5.487890515478721, 6.248593623100655, 6.658277973360080, 7.271761285327994, 7.366181860954229, 8.395251545980919, 8.482519853553075, 9.059850001737780, 9.372060472051871, 10.07204244849483, 10.46401546582762, 10.70500806944401, 11.33054992250521, 11.77251382875248, 12.62035376371012, 12.74673430205200

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