Properties

Label 2-1008-1.1-c5-0-18
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  + 34·5-s − 49·7-s − 756·11-s + 678·13-s + 1.83e3·17-s − 604·19-s + 2.84e3·23-s − 1.96e3·25-s − 6.87e3·29-s − 3.56e3·31-s − 1.66e3·35-s + 1.45e4·37-s − 5.96e3·41-s + 676·43-s − 2.08e4·47-s + 2.40e3·49-s − 3.23e4·53-s − 2.57e4·55-s + 4.29e4·59-s + 4.48e4·61-s + 2.30e4·65-s + 3.97e4·67-s − 2.58e4·71-s + 5.89e4·73-s + 3.70e4·77-s + 7.76e4·79-s + 3.59e4·83-s + ⋯
L(s)  = 1  + 0.608·5-s − 0.377·7-s − 1.88·11-s + 1.11·13-s + 1.54·17-s − 0.383·19-s + 1.11·23-s − 0.630·25-s − 1.51·29-s − 0.666·31-s − 0.229·35-s + 1.75·37-s − 0.553·41-s + 0.0557·43-s − 1.37·47-s + 1/7·49-s − 1.58·53-s − 1.14·55-s + 1.60·59-s + 1.54·61-s + 0.676·65-s + 1.08·67-s − 0.607·71-s + 1.29·73-s + 0.712·77-s + 1.39·79-s + 0.573·83-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

\(L(3)\) \(\approx\) \(2.077352082\)
\(L(\frac12)\) \(\approx\) \(2.077352082\)
\(L(\frac{7}{2})\) 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 \( 1 - 34 T + p^{5} T^{2} \)
11 \( 1 + 756 T + p^{5} T^{2} \)
13 \( 1 - 678 T + p^{5} T^{2} \)
17 \( 1 - 1838 T + p^{5} T^{2} \)
19 \( 1 + 604 T + p^{5} T^{2} \)
23 \( 1 - 2840 T + p^{5} T^{2} \)
29 \( 1 + 6878 T + p^{5} T^{2} \)
31 \( 1 + 3568 T + p^{5} T^{2} \)
37 \( 1 - 14598 T + p^{5} T^{2} \)
41 \( 1 + 5962 T + p^{5} T^{2} \)
43 \( 1 - 676 T + p^{5} T^{2} \)
47 \( 1 + 20800 T + p^{5} T^{2} \)
53 \( 1 + 32390 T + p^{5} T^{2} \)
59 \( 1 - 42948 T + p^{5} T^{2} \)
61 \( 1 - 44806 T + p^{5} T^{2} \)
67 \( 1 - 39708 T + p^{5} T^{2} \)
71 \( 1 + 25800 T + p^{5} T^{2} \)
73 \( 1 - 58954 T + p^{5} T^{2} \)
79 \( 1 - 77648 T + p^{5} T^{2} \)
83 \( 1 - 35964 T + p^{5} T^{2} \)
89 \( 1 + 80842 T + p^{5} T^{2} \)
97 \( 1 + 64334 T + p^{5} T^{2} \)
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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

−9.457662962186780430207734132965, −8.231773993539421759352580221006, −7.74283857043071508629398208364, −6.61954142161766780943140301527, −5.61579149364430341583128272115, −5.25250922574858979447720879021, −3.75362964182239988900417200560, −2.90599544726553754262894471920, −1.84793339102824053792151894508, −0.61962131237297326531883162655, 0.61962131237297326531883162655, 1.84793339102824053792151894508, 2.90599544726553754262894471920, 3.75362964182239988900417200560, 5.25250922574858979447720879021, 5.61579149364430341583128272115, 6.61954142161766780943140301527, 7.74283857043071508629398208364, 8.231773993539421759352580221006, 9.457662962186780430207734132965

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