Properties

Label 2-1008-1.1-c3-0-10
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 + 36·11-s + 62·13-s − 114·17-s + 76·19-s − 24·23-s − 89·25-s − 54·29-s + 112·31-s + 42·35-s − 178·37-s − 378·41-s + 172·43-s − 192·47-s + 49·49-s + 402·53-s − 216·55-s + 396·59-s + 254·61-s − 372·65-s + 1.01e3·67-s + 840·71-s + 890·73-s − 252·77-s − 80·79-s − 108·83-s + ⋯
L(s)  = 1  − 0.536·5-s − 0.377·7-s + 0.986·11-s + 1.32·13-s − 1.62·17-s + 0.917·19-s − 0.217·23-s − 0.711·25-s − 0.345·29-s + 0.648·31-s + 0.202·35-s − 0.790·37-s − 1.43·41-s + 0.609·43-s − 0.595·47-s + 1/7·49-s + 1.04·53-s − 0.529·55-s + 0.873·59-s + 0.533·61-s − 0.709·65-s + 1.84·67-s + 1.40·71-s + 1.42·73-s − 0.372·77-s − 0.113·79-s − 0.142·83-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

\(L(2)\) \(\approx\) \(1.763769769\)
\(L(\frac12)\) \(\approx\) \(1.763769769\)
\(L(\frac{5}{2})\) 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 \( 1 - 36 T + p^{3} T^{2} \)
13 \( 1 - 62 T + p^{3} T^{2} \)
17 \( 1 + 114 T + p^{3} T^{2} \)
19 \( 1 - 4 p T + p^{3} T^{2} \)
23 \( 1 + 24 T + p^{3} T^{2} \)
29 \( 1 + 54 T + p^{3} T^{2} \)
31 \( 1 - 112 T + p^{3} T^{2} \)
37 \( 1 + 178 T + p^{3} T^{2} \)
41 \( 1 + 378 T + p^{3} T^{2} \)
43 \( 1 - 4 p T + p^{3} T^{2} \)
47 \( 1 + 192 T + p^{3} T^{2} \)
53 \( 1 - 402 T + p^{3} T^{2} \)
59 \( 1 - 396 T + p^{3} T^{2} \)
61 \( 1 - 254 T + p^{3} T^{2} \)
67 \( 1 - 1012 T + p^{3} T^{2} \)
71 \( 1 - 840 T + p^{3} T^{2} \)
73 \( 1 - 890 T + p^{3} T^{2} \)
79 \( 1 + 80 T + p^{3} T^{2} \)
83 \( 1 + 108 T + p^{3} T^{2} \)
89 \( 1 - 1638 T + p^{3} T^{2} \)
97 \( 1 - 1010 T + p^{3} T^{2} \)
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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

−9.462721951179014409369429757499, −8.759532632335360220724735244522, −8.046420865894661170600076981122, −6.85035290503471266410248357350, −6.40799379737590035443009070372, −5.24858383371421549637235267853, −4.00937605471572853659508677316, −3.53188403676786521693653133688, −2.02226335537543742977171335832, −0.71218789596236189887666092513, 0.71218789596236189887666092513, 2.02226335537543742977171335832, 3.53188403676786521693653133688, 4.00937605471572853659508677316, 5.24858383371421549637235267853, 6.40799379737590035443009070372, 6.85035290503471266410248357350, 8.046420865894661170600076981122, 8.759532632335360220724735244522, 9.462721951179014409369429757499

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