Properties

Label 2-1008-1.1-c5-0-39
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 81.4·5-s − 49·7-s + 340.·11-s − 1.10e3·13-s − 197.·17-s + 2.33e3·19-s − 2.60e3·23-s + 3.50e3·25-s + 7.91e3·29-s + 9.04e3·31-s + 3.98e3·35-s − 5.47e3·37-s − 1.52e4·41-s + 3.82e3·43-s − 1.94e3·47-s + 2.40e3·49-s + 2.62e4·53-s − 2.77e4·55-s + 4.58e4·59-s − 4.34e4·61-s + 8.96e4·65-s − 1.58e4·67-s + 2.41e4·71-s + 6.90e4·73-s − 1.66e4·77-s − 6.09e4·79-s − 5.22e4·83-s + ⋯
L(s)  = 1  − 1.45·5-s − 0.377·7-s + 0.847·11-s − 1.80·13-s − 0.165·17-s + 1.48·19-s − 1.02·23-s + 1.12·25-s + 1.74·29-s + 1.69·31-s + 0.550·35-s − 0.657·37-s − 1.41·41-s + 0.315·43-s − 0.128·47-s + 0.142·49-s + 1.28·53-s − 1.23·55-s + 1.71·59-s − 1.49·61-s + 2.63·65-s − 0.431·67-s + 0.567·71-s + 1.51·73-s − 0.320·77-s − 1.09·79-s − 0.832·83-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: \(1\)
Selberg data: \((2,\ 1008,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 81.4T + 3.12e3T^{2} \)
11 \( 1 - 340.T + 1.61e5T^{2} \)
13 \( 1 + 1.10e3T + 3.71e5T^{2} \)
17 \( 1 + 197.T + 1.41e6T^{2} \)
19 \( 1 - 2.33e3T + 2.47e6T^{2} \)
23 \( 1 + 2.60e3T + 6.43e6T^{2} \)
29 \( 1 - 7.91e3T + 2.05e7T^{2} \)
31 \( 1 - 9.04e3T + 2.86e7T^{2} \)
37 \( 1 + 5.47e3T + 6.93e7T^{2} \)
41 \( 1 + 1.52e4T + 1.15e8T^{2} \)
43 \( 1 - 3.82e3T + 1.47e8T^{2} \)
47 \( 1 + 1.94e3T + 2.29e8T^{2} \)
53 \( 1 - 2.62e4T + 4.18e8T^{2} \)
59 \( 1 - 4.58e4T + 7.14e8T^{2} \)
61 \( 1 + 4.34e4T + 8.44e8T^{2} \)
67 \( 1 + 1.58e4T + 1.35e9T^{2} \)
71 \( 1 - 2.41e4T + 1.80e9T^{2} \)
73 \( 1 - 6.90e4T + 2.07e9T^{2} \)
79 \( 1 + 6.09e4T + 3.07e9T^{2} \)
83 \( 1 + 5.22e4T + 3.93e9T^{2} \)
89 \( 1 - 1.00e5T + 5.58e9T^{2} \)
97 \( 1 - 6.49e4T + 8.58e9T^{2} \)
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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

−8.691081174672342689387434617666, −7.922709120305317669816824655712, −7.18795758970290832053440849532, −6.52401463942860405473637564265, −5.12616711312231625208631785987, −4.37740730790668862888850354649, −3.47627810818129741369723508827, −2.56138344703063950124839255276, −0.958003425478879581946950340638, 0, 0.958003425478879581946950340638, 2.56138344703063950124839255276, 3.47627810818129741369723508827, 4.37740730790668862888850354649, 5.12616711312231625208631785987, 6.52401463942860405473637564265, 7.18795758970290832053440849532, 7.922709120305317669816824655712, 8.691081174672342689387434617666

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