Properties

Label 2-832-1.1-c3-0-32
Degree $2$
Conductor $832$
Sign $1$
Analytic cond. $49.0895$
Root an. cond. $7.00639$
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  − 4·3-s + 18·5-s + 20·7-s − 11·9-s + 48·11-s − 13·13-s − 72·15-s + 66·17-s + 16·19-s − 80·21-s + 168·23-s + 199·25-s + 152·27-s − 6·29-s + 20·31-s − 192·33-s + 360·35-s − 254·37-s + 52·39-s − 390·41-s + 124·43-s − 198·45-s − 468·47-s + 57·49-s − 264·51-s − 558·53-s + 864·55-s + ⋯
L(s)  = 1  − 0.769·3-s + 1.60·5-s + 1.07·7-s − 0.407·9-s + 1.31·11-s − 0.277·13-s − 1.23·15-s + 0.941·17-s + 0.193·19-s − 0.831·21-s + 1.52·23-s + 1.59·25-s + 1.08·27-s − 0.0384·29-s + 0.115·31-s − 1.01·33-s + 1.73·35-s − 1.12·37-s + 0.213·39-s − 1.48·41-s + 0.439·43-s − 0.655·45-s − 1.45·47-s + 0.166·49-s − 0.724·51-s − 1.44·53-s + 2.11·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $1$
Analytic conductor: \(49.0895\)
Root analytic conductor: \(7.00639\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 832,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.737908646\)
\(L(\frac12)\) \(\approx\) \(2.737908646\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + p T \)
good3 \( 1 + 4 T + p^{3} T^{2} \)
5 \( 1 - 18 T + p^{3} T^{2} \)
7 \( 1 - 20 T + p^{3} T^{2} \)
11 \( 1 - 48 T + p^{3} T^{2} \)
17 \( 1 - 66 T + p^{3} T^{2} \)
19 \( 1 - 16 T + p^{3} T^{2} \)
23 \( 1 - 168 T + p^{3} T^{2} \)
29 \( 1 + 6 T + p^{3} T^{2} \)
31 \( 1 - 20 T + p^{3} T^{2} \)
37 \( 1 + 254 T + p^{3} T^{2} \)
41 \( 1 + 390 T + p^{3} T^{2} \)
43 \( 1 - 124 T + p^{3} T^{2} \)
47 \( 1 + 468 T + p^{3} T^{2} \)
53 \( 1 + 558 T + p^{3} T^{2} \)
59 \( 1 - 96 T + p^{3} T^{2} \)
61 \( 1 - 826 T + p^{3} T^{2} \)
67 \( 1 - 160 T + p^{3} T^{2} \)
71 \( 1 + 420 T + p^{3} T^{2} \)
73 \( 1 - 362 T + p^{3} T^{2} \)
79 \( 1 - 776 T + p^{3} T^{2} \)
83 \( 1 + p^{3} T^{2} \)
89 \( 1 - 1626 T + p^{3} T^{2} \)
97 \( 1 + 1294 T + p^{3} T^{2} \)
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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

−9.840440748898604325462791342154, −9.110861677952599689750735459485, −8.278749832917521996793566724722, −6.91783550436518644219459408404, −6.28313943330232167553960945462, −5.27552564752446887702134659181, −4.96392201530735620986206959396, −3.25450287395105571139907705422, −1.84794705520732562701466505448, −1.05826629559066423815847739094, 1.05826629559066423815847739094, 1.84794705520732562701466505448, 3.25450287395105571139907705422, 4.96392201530735620986206959396, 5.27552564752446887702134659181, 6.28313943330232167553960945462, 6.91783550436518644219459408404, 8.278749832917521996793566724722, 9.110861677952599689750735459485, 9.840440748898604325462791342154

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