Properties

Label 2-1008-1.1-c5-0-7
Degree $2$
Conductor $1008$
Sign $1$
Analytic cond. $161.666$
Root an. cond. $12.7148$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 71.6·5-s + 49·7-s − 567.·11-s + 831.·13-s − 888.·17-s − 2.91e3·19-s + 3.10e3·23-s + 2.00e3·25-s − 8.27e3·29-s + 7.02e3·31-s − 3.50e3·35-s − 1.01e4·37-s − 3.09e3·41-s − 1.50e4·43-s + 1.98e4·47-s + 2.40e3·49-s + 9.20e3·53-s + 4.06e4·55-s − 1.03e4·59-s − 2.25e4·61-s − 5.95e4·65-s − 6.41e3·67-s − 6.12e4·71-s − 2.97e4·73-s − 2.78e4·77-s + 1.56e4·79-s + 1.66e3·83-s + ⋯
L(s)  = 1  − 1.28·5-s + 0.377·7-s − 1.41·11-s + 1.36·13-s − 0.745·17-s − 1.85·19-s + 1.22·23-s + 0.641·25-s − 1.82·29-s + 1.31·31-s − 0.484·35-s − 1.21·37-s − 0.287·41-s − 1.23·43-s + 1.31·47-s + 0.142·49-s + 0.450·53-s + 1.81·55-s − 0.385·59-s − 0.777·61-s − 1.74·65-s − 0.174·67-s − 1.44·71-s − 0.652·73-s − 0.534·77-s + 0.281·79-s + 0.0265·83-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7610038303\)
\(L(\frac12)\) \(\approx\) \(0.7610038303\)
\(L(\frac{7}{2})\) 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 \( 1 - 49T \)
good5 \( 1 + 71.6T + 3.12e3T^{2} \)
11 \( 1 + 567.T + 1.61e5T^{2} \)
13 \( 1 - 831.T + 3.71e5T^{2} \)
17 \( 1 + 888.T + 1.41e6T^{2} \)
19 \( 1 + 2.91e3T + 2.47e6T^{2} \)
23 \( 1 - 3.10e3T + 6.43e6T^{2} \)
29 \( 1 + 8.27e3T + 2.05e7T^{2} \)
31 \( 1 - 7.02e3T + 2.86e7T^{2} \)
37 \( 1 + 1.01e4T + 6.93e7T^{2} \)
41 \( 1 + 3.09e3T + 1.15e8T^{2} \)
43 \( 1 + 1.50e4T + 1.47e8T^{2} \)
47 \( 1 - 1.98e4T + 2.29e8T^{2} \)
53 \( 1 - 9.20e3T + 4.18e8T^{2} \)
59 \( 1 + 1.03e4T + 7.14e8T^{2} \)
61 \( 1 + 2.25e4T + 8.44e8T^{2} \)
67 \( 1 + 6.41e3T + 1.35e9T^{2} \)
71 \( 1 + 6.12e4T + 1.80e9T^{2} \)
73 \( 1 + 2.97e4T + 2.07e9T^{2} \)
79 \( 1 - 1.56e4T + 3.07e9T^{2} \)
83 \( 1 - 1.66e3T + 3.93e9T^{2} \)
89 \( 1 + 7.58e4T + 5.58e9T^{2} \)
97 \( 1 + 9.80e4T + 8.58e9T^{2} \)
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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

−8.788304226857600252919847954097, −8.496691580693985168973122679887, −7.65559443042212896723051636877, −6.85115370563623636349413323482, −5.78491089347295753035898415199, −4.71510774637363447902838894519, −3.99730444694821611228253978938, −3.01401410519866040547444543657, −1.80166700104144915074050944718, −0.36925402850570488271511580669, 0.36925402850570488271511580669, 1.80166700104144915074050944718, 3.01401410519866040547444543657, 3.99730444694821611228253978938, 4.71510774637363447902838894519, 5.78491089347295753035898415199, 6.85115370563623636349413323482, 7.65559443042212896723051636877, 8.496691580693985168973122679887, 8.788304226857600252919847954097

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