Properties

Label 2-1008-1.1-c5-0-32
Degree $2$
Conductor $1008$
Sign $1$
Analytic cond. $161.666$
Root an. cond. $12.7148$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 56·5-s + 49·7-s + 232·11-s − 140·13-s + 1.72e3·17-s + 98·19-s + 1.82e3·23-s + 11·25-s − 3.41e3·29-s + 7.64e3·31-s + 2.74e3·35-s − 1.03e4·37-s + 1.79e4·41-s − 1.08e4·43-s + 9.32e3·47-s + 2.40e3·49-s − 2.26e3·53-s + 1.29e4·55-s − 2.73e3·59-s + 2.56e4·61-s − 7.84e3·65-s + 4.84e4·67-s − 5.85e4·71-s + 6.80e4·73-s + 1.13e4·77-s − 3.17e4·79-s − 2.05e4·83-s + ⋯
L(s)  = 1  + 1.00·5-s + 0.377·7-s + 0.578·11-s − 0.229·13-s + 1.44·17-s + 0.0622·19-s + 0.718·23-s + 0.00351·25-s − 0.754·29-s + 1.42·31-s + 0.378·35-s − 1.24·37-s + 1.66·41-s − 0.897·43-s + 0.615·47-s + 1/7·49-s − 0.110·53-s + 0.579·55-s − 0.102·59-s + 0.882·61-s − 0.230·65-s + 1.31·67-s − 1.37·71-s + 1.49·73-s + 0.218·77-s − 0.572·79-s − 0.327·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(161.666\)
Root analytic conductor: \(12.7148\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.524119855\)
\(L(\frac12)\) \(\approx\) \(3.524119855\)
\(L(\frac{7}{2})\) 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 \( 1 \)
7 \( 1 - p^{2} T \)
good5 \( 1 - 56 T + p^{5} T^{2} \)
11 \( 1 - 232 T + p^{5} T^{2} \)
13 \( 1 + 140 T + p^{5} T^{2} \)
17 \( 1 - 1722 T + p^{5} T^{2} \)
19 \( 1 - 98 T + p^{5} T^{2} \)
23 \( 1 - 1824 T + p^{5} T^{2} \)
29 \( 1 + 3418 T + p^{5} T^{2} \)
31 \( 1 - 7644 T + p^{5} T^{2} \)
37 \( 1 + 10398 T + p^{5} T^{2} \)
41 \( 1 - 17962 T + p^{5} T^{2} \)
43 \( 1 + 10880 T + p^{5} T^{2} \)
47 \( 1 - 9324 T + p^{5} T^{2} \)
53 \( 1 + 2262 T + p^{5} T^{2} \)
59 \( 1 + 2730 T + p^{5} T^{2} \)
61 \( 1 - 25648 T + p^{5} T^{2} \)
67 \( 1 - 48404 T + p^{5} T^{2} \)
71 \( 1 + 58560 T + p^{5} T^{2} \)
73 \( 1 - 68082 T + p^{5} T^{2} \)
79 \( 1 + 31784 T + p^{5} T^{2} \)
83 \( 1 + 20538 T + p^{5} T^{2} \)
89 \( 1 - 50582 T + p^{5} T^{2} \)
97 \( 1 + 58506 T + p^{5} T^{2} \)
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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

−9.398477043642427168506661127202, −8.451343349663829261179473281920, −7.55389795816327604446994325656, −6.64128478356450729501292710811, −5.72931648971684969204398740727, −5.09563746154276091787362263790, −3.91152253999964952690799155910, −2.80362894765330056020242029140, −1.73110963842308953837118532936, −0.868197551794566994645427025971, 0.868197551794566994645427025971, 1.73110963842308953837118532936, 2.80362894765330056020242029140, 3.91152253999964952690799155910, 5.09563746154276091787362263790, 5.72931648971684969204398740727, 6.64128478356450729501292710811, 7.55389795816327604446994325656, 8.451343349663829261179473281920, 9.398477043642427168506661127202

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