Properties

Label 2-882-1.1-c3-0-12
Degree $2$
Conductor $882$
Sign $1$
Analytic cond. $52.0396$
Root an. cond. $7.21385$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·4-s − 7·5-s + 8·8-s − 14·10-s − 35·11-s + 66·13-s + 16·16-s − 59·17-s + 137·19-s − 28·20-s − 70·22-s + 7·23-s − 76·25-s + 132·26-s − 106·29-s + 75·31-s + 32·32-s − 118·34-s + 11·37-s + 274·38-s − 56·40-s + 498·41-s + 260·43-s − 140·44-s + 14·46-s + 171·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.626·5-s + 0.353·8-s − 0.442·10-s − 0.959·11-s + 1.40·13-s + 1/4·16-s − 0.841·17-s + 1.65·19-s − 0.313·20-s − 0.678·22-s + 0.0634·23-s − 0.607·25-s + 0.995·26-s − 0.678·29-s + 0.434·31-s + 0.176·32-s − 0.595·34-s + 0.0488·37-s + 1.16·38-s − 0.221·40-s + 1.89·41-s + 0.922·43-s − 0.479·44-s + 0.0448·46-s + 0.530·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(52.0396\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.919842764\)
\(L(\frac12)\) \(\approx\) \(2.919842764\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 7 T + p^{3} T^{2} \)
11 \( 1 + 35 T + p^{3} T^{2} \)
13 \( 1 - 66 T + p^{3} T^{2} \)
17 \( 1 + 59 T + p^{3} T^{2} \)
19 \( 1 - 137 T + p^{3} T^{2} \)
23 \( 1 - 7 T + p^{3} T^{2} \)
29 \( 1 + 106 T + p^{3} T^{2} \)
31 \( 1 - 75 T + p^{3} T^{2} \)
37 \( 1 - 11 T + p^{3} T^{2} \)
41 \( 1 - 498 T + p^{3} T^{2} \)
43 \( 1 - 260 T + p^{3} T^{2} \)
47 \( 1 - 171 T + p^{3} T^{2} \)
53 \( 1 - 417 T + p^{3} T^{2} \)
59 \( 1 - 17 T + p^{3} T^{2} \)
61 \( 1 - 51 T + p^{3} T^{2} \)
67 \( 1 - 439 T + p^{3} T^{2} \)
71 \( 1 - 784 T + p^{3} T^{2} \)
73 \( 1 - 295 T + p^{3} T^{2} \)
79 \( 1 + 495 T + p^{3} T^{2} \)
83 \( 1 + 932 T + p^{3} T^{2} \)
89 \( 1 - 873 T + p^{3} T^{2} \)
97 \( 1 + 290 T + p^{3} T^{2} \)
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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

−9.832772566629396652293579695876, −8.814501140936942194141506730156, −7.85660338224588684498842615893, −7.26518799774377905145726232697, −6.08207830556770713062919543836, −5.38407292387238368362552907194, −4.25703453929847166151377867060, −3.49203308904116959308369358584, −2.39289872844953941895722807491, −0.852502094756500337920806554401, 0.852502094756500337920806554401, 2.39289872844953941895722807491, 3.49203308904116959308369358584, 4.25703453929847166151377867060, 5.38407292387238368362552907194, 6.08207830556770713062919543836, 7.26518799774377905145726232697, 7.85660338224588684498842615893, 8.814501140936942194141506730156, 9.832772566629396652293579695876

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