Properties

Label 2-1008-1.1-c5-0-6
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  − 94·5-s + 49·7-s + 52·11-s − 770·13-s + 2.02e3·17-s − 1.73e3·19-s − 576·23-s + 5.71e3·25-s − 5.51e3·29-s − 6.33e3·31-s − 4.60e3·35-s − 7.33e3·37-s + 3.26e3·41-s − 5.42e3·43-s + 864·47-s + 2.40e3·49-s − 4.18e3·53-s − 4.88e3·55-s − 1.12e4·59-s − 4.56e4·61-s + 7.23e4·65-s − 1.39e3·67-s + 1.87e4·71-s + 4.63e4·73-s + 2.54e3·77-s − 9.74e4·79-s − 8.12e4·83-s + ⋯
L(s)  = 1  − 1.68·5-s + 0.377·7-s + 0.129·11-s − 1.26·13-s + 1.69·17-s − 1.10·19-s − 0.227·23-s + 1.82·25-s − 1.21·29-s − 1.18·31-s − 0.635·35-s − 0.881·37-s + 0.303·41-s − 0.447·43-s + 0.0570·47-s + 1/7·49-s − 0.204·53-s − 0.217·55-s − 0.419·59-s − 1.56·61-s + 2.12·65-s − 0.0379·67-s + 0.440·71-s + 1.01·73-s + 0.0489·77-s − 1.75·79-s − 1.29·83-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

\(L(3)\) \(\approx\) \(0.6852124112\)
\(L(\frac12)\) \(\approx\) \(0.6852124112\)
\(L(\frac{7}{2})\) 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 \( 1 + 94 T + p^{5} T^{2} \)
11 \( 1 - 52 T + p^{5} T^{2} \)
13 \( 1 + 770 T + p^{5} T^{2} \)
17 \( 1 - 2022 T + p^{5} T^{2} \)
19 \( 1 + 1732 T + p^{5} T^{2} \)
23 \( 1 + 576 T + p^{5} T^{2} \)
29 \( 1 + 5518 T + p^{5} T^{2} \)
31 \( 1 + 6336 T + p^{5} T^{2} \)
37 \( 1 + 7338 T + p^{5} T^{2} \)
41 \( 1 - 3262 T + p^{5} T^{2} \)
43 \( 1 + 5420 T + p^{5} T^{2} \)
47 \( 1 - 864 T + p^{5} T^{2} \)
53 \( 1 + 4182 T + p^{5} T^{2} \)
59 \( 1 + 11220 T + p^{5} T^{2} \)
61 \( 1 + 45602 T + p^{5} T^{2} \)
67 \( 1 + 1396 T + p^{5} T^{2} \)
71 \( 1 - 18720 T + p^{5} T^{2} \)
73 \( 1 - 46362 T + p^{5} T^{2} \)
79 \( 1 + 97424 T + p^{5} T^{2} \)
83 \( 1 + 81228 T + p^{5} T^{2} \)
89 \( 1 - 3182 T + p^{5} T^{2} \)
97 \( 1 - 4914 T + p^{5} T^{2} \)
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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

−9.125730741282897759883330918300, −8.192181256962075934278441833806, −7.58571814611712802713661660971, −7.07444327717585530945671090684, −5.67960752359801247624129043940, −4.75046015623556923209011072046, −3.91950566843777944807525440828, −3.12659339742169062583788704767, −1.74648622040631876766870694834, −0.35451076730960464996350421293, 0.35451076730960464996350421293, 1.74648622040631876766870694834, 3.12659339742169062583788704767, 3.91950566843777944807525440828, 4.75046015623556923209011072046, 5.67960752359801247624129043940, 7.07444327717585530945671090684, 7.58571814611712802713661660971, 8.192181256962075934278441833806, 9.125730741282897759883330918300

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