Properties

Label 2-1008-1.1-c3-0-4
Degree $2$
Conductor $1008$
Sign $1$
Analytic cond. $59.4739$
Root an. cond. $7.71193$
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  − 6·5-s − 7·7-s − 12·11-s − 82·13-s + 30·17-s − 68·19-s + 216·23-s − 89·25-s − 246·29-s + 112·31-s + 42·35-s + 110·37-s + 246·41-s + 172·43-s + 192·47-s + 49·49-s − 558·53-s + 72·55-s + 540·59-s + 110·61-s + 492·65-s − 140·67-s − 840·71-s − 550·73-s + 84·77-s + 208·79-s + 516·83-s + ⋯
L(s)  = 1  − 0.536·5-s − 0.377·7-s − 0.328·11-s − 1.74·13-s + 0.428·17-s − 0.821·19-s + 1.95·23-s − 0.711·25-s − 1.57·29-s + 0.648·31-s + 0.202·35-s + 0.488·37-s + 0.937·41-s + 0.609·43-s + 0.595·47-s + 1/7·49-s − 1.44·53-s + 0.176·55-s + 1.19·59-s + 0.230·61-s + 0.938·65-s − 0.255·67-s − 1.40·71-s − 0.881·73-s + 0.124·77-s + 0.296·79-s + 0.682·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(59.4739\)
Root analytic conductor: \(7.71193\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.122983011\)
\(L(\frac12)\) \(\approx\) \(1.122983011\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + p T \)
good5 \( 1 + 6 T + p^{3} T^{2} \)
11 \( 1 + 12 T + p^{3} T^{2} \)
13 \( 1 + 82 T + p^{3} T^{2} \)
17 \( 1 - 30 T + p^{3} T^{2} \)
19 \( 1 + 68 T + p^{3} T^{2} \)
23 \( 1 - 216 T + p^{3} T^{2} \)
29 \( 1 + 246 T + p^{3} T^{2} \)
31 \( 1 - 112 T + p^{3} T^{2} \)
37 \( 1 - 110 T + p^{3} T^{2} \)
41 \( 1 - 6 p T + p^{3} T^{2} \)
43 \( 1 - 4 p T + p^{3} T^{2} \)
47 \( 1 - 192 T + p^{3} T^{2} \)
53 \( 1 + 558 T + p^{3} T^{2} \)
59 \( 1 - 540 T + p^{3} T^{2} \)
61 \( 1 - 110 T + p^{3} T^{2} \)
67 \( 1 + 140 T + p^{3} T^{2} \)
71 \( 1 + 840 T + p^{3} T^{2} \)
73 \( 1 + 550 T + p^{3} T^{2} \)
79 \( 1 - 208 T + p^{3} T^{2} \)
83 \( 1 - 516 T + p^{3} T^{2} \)
89 \( 1 - 1398 T + p^{3} T^{2} \)
97 \( 1 - 1586 T + p^{3} T^{2} \)
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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

−9.555306773507720860577488870593, −8.867913280505902025326905783237, −7.59382569275539513688835172809, −7.38355710508562751144968135554, −6.18426148452999220828357010451, −5.15539095999123937951856941625, −4.33768494447444029632537673545, −3.18293892443071277236759996534, −2.23326118303487741740677473394, −0.53933110866247915680393449368, 0.53933110866247915680393449368, 2.23326118303487741740677473394, 3.18293892443071277236759996534, 4.33768494447444029632537673545, 5.15539095999123937951856941625, 6.18426148452999220828357010451, 7.38355710508562751144968135554, 7.59382569275539513688835172809, 8.867913280505902025326905783237, 9.555306773507720860577488870593

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