Properties

Label 2-912-1.1-c1-0-7
Degree $2$
Conductor $912$
Sign $1$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
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 + 2·5-s + 9-s + 4·11-s + 2·13-s + 2·15-s − 6·17-s + 19-s + 4·23-s − 25-s + 27-s − 2·29-s − 4·31-s + 4·33-s + 10·37-s + 2·39-s + 10·41-s − 4·43-s + 2·45-s + 4·47-s − 7·49-s − 6·51-s − 10·53-s + 8·55-s + 57-s − 12·59-s + 14·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.516·15-s − 1.45·17-s + 0.229·19-s + 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.696·33-s + 1.64·37-s + 0.320·39-s + 1.56·41-s − 0.609·43-s + 0.298·45-s + 0.583·47-s − 49-s − 0.840·51-s − 1.37·53-s + 1.07·55-s + 0.132·57-s − 1.56·59-s + 1.79·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $1$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{912} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.388472434\)
\(L(\frac12)\) \(\approx\) \(2.388472434\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
19 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−9.786869452003372550002449800084, −9.259741991745774642651687902060, −8.686112233403729699672924385749, −7.54920325839787930805373575681, −6.57061735757055000239038218298, −5.95928356394221873345303919204, −4.65672229415455481414222119791, −3.72816659383691276241941645312, −2.47746893642927435624182028712, −1.41228664440352179477085709942, 1.41228664440352179477085709942, 2.47746893642927435624182028712, 3.72816659383691276241941645312, 4.65672229415455481414222119791, 5.95928356394221873345303919204, 6.57061735757055000239038218298, 7.54920325839787930805373575681, 8.686112233403729699672924385749, 9.259741991745774642651687902060, 9.786869452003372550002449800084

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