Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 1.20·5-s − 1.61·7-s + 9-s + 0.338·11-s + 3.11·13-s − 1.20·15-s − 0.639·17-s + 0.596·19-s + 1.61·21-s − 23-s − 3.55·25-s − 27-s + 29-s + 6.81·31-s − 0.338·33-s − 1.93·35-s − 9.39·37-s − 3.11·39-s − 7.25·41-s − 10.1·43-s + 1.20·45-s + 8.35·47-s − 4.39·49-s + 0.639·51-s − 3.58·53-s + 0.406·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.537·5-s − 0.609·7-s + 0.333·9-s + 0.101·11-s + 0.862·13-s − 0.310·15-s − 0.155·17-s + 0.136·19-s + 0.351·21-s − 0.208·23-s − 0.711·25-s − 0.192·27-s + 0.185·29-s + 1.22·31-s − 0.0588·33-s − 0.327·35-s − 1.54·37-s − 0.498·39-s − 1.13·41-s − 1.54·43-s + 0.179·45-s + 1.21·47-s − 0.628·49-s + 0.0894·51-s − 0.492·53-s + 0.0547·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8004\)    =    \(2^{2} \cdot 3 \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8004} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 8004,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;23,\;29\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;23,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
good5 \( 1 - 1.20T + 5T^{2} \)
7 \( 1 + 1.61T + 7T^{2} \)
11 \( 1 - 0.338T + 11T^{2} \)
13 \( 1 - 3.11T + 13T^{2} \)
17 \( 1 + 0.639T + 17T^{2} \)
19 \( 1 - 0.596T + 19T^{2} \)
31 \( 1 - 6.81T + 31T^{2} \)
37 \( 1 + 9.39T + 37T^{2} \)
41 \( 1 + 7.25T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 - 8.35T + 47T^{2} \)
53 \( 1 + 3.58T + 53T^{2} \)
59 \( 1 + 7.07T + 59T^{2} \)
61 \( 1 - 2.15T + 61T^{2} \)
67 \( 1 - 14.3T + 67T^{2} \)
71 \( 1 - 2.94T + 71T^{2} \)
73 \( 1 + 2.80T + 73T^{2} \)
79 \( 1 + 0.0498T + 79T^{2} \)
83 \( 1 + 1.56T + 83T^{2} \)
89 \( 1 - 1.00T + 89T^{2} \)
97 \( 1 - 6.80T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.33718381552227437753192017555, −6.49884263371389857118428022857, −6.30101355173058507574180777541, −5.44260844574766274456896887158, −4.82593836126246172389436066374, −3.83553361247895264641681536758, −3.22204000997571439855715330273, −2.10294238094845670970731899233, −1.24680424123389071675228460127, 0, 1.24680424123389071675228460127, 2.10294238094845670970731899233, 3.22204000997571439855715330273, 3.83553361247895264641681536758, 4.82593836126246172389436066374, 5.44260844574766274456896887158, 6.30101355173058507574180777541, 6.49884263371389857118428022857, 7.33718381552227437753192017555

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