Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 1.63·5-s − 0.577·7-s + 9-s + 5.51·11-s + 0.530·13-s + 1.63·15-s + 6.30·17-s + 0.869·19-s − 0.577·21-s + 23-s − 2.34·25-s + 27-s − 29-s − 4.68·31-s + 5.51·33-s − 0.941·35-s − 2.51·37-s + 0.530·39-s + 7.33·41-s + 7.82·43-s + 1.63·45-s + 4.94·47-s − 6.66·49-s + 6.30·51-s + 2.53·53-s + 8.99·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.728·5-s − 0.218·7-s + 0.333·9-s + 1.66·11-s + 0.147·13-s + 0.420·15-s + 1.52·17-s + 0.199·19-s − 0.126·21-s + 0.208·23-s − 0.468·25-s + 0.192·27-s − 0.185·29-s − 0.840·31-s + 0.960·33-s − 0.159·35-s − 0.414·37-s + 0.0848·39-s + 1.14·41-s + 1.19·43-s + 0.242·45-s + 0.721·47-s − 0.952·49-s + 0.883·51-s + 0.348·53-s + 1.21·55-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  =  0
Selberg data  =  $(2,\ 8004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.719388802$
$L(\frac12)$  $\approx$  $3.719388802$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 + T \)
good5 \( 1 - 1.63T + 5T^{2} \)
7 \( 1 + 0.577T + 7T^{2} \)
11 \( 1 - 5.51T + 11T^{2} \)
13 \( 1 - 0.530T + 13T^{2} \)
17 \( 1 - 6.30T + 17T^{2} \)
19 \( 1 - 0.869T + 19T^{2} \)
31 \( 1 + 4.68T + 31T^{2} \)
37 \( 1 + 2.51T + 37T^{2} \)
41 \( 1 - 7.33T + 41T^{2} \)
43 \( 1 - 7.82T + 43T^{2} \)
47 \( 1 - 4.94T + 47T^{2} \)
53 \( 1 - 2.53T + 53T^{2} \)
59 \( 1 - 14.0T + 59T^{2} \)
61 \( 1 + 5.51T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 + 9.56T + 71T^{2} \)
73 \( 1 + 3.25T + 73T^{2} \)
79 \( 1 - 0.347T + 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 - 2.03T + 97T^{2} \)
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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

−7.64056190935897453511080817401, −7.33281687794757748950948333520, −6.30696350075628989789053458923, −5.92686981145048990162068997783, −5.10924164426307824879037252386, −4.00806065561990963619132795847, −3.60757675599384505485592016635, −2.67917047695031923373062108827, −1.70764937931677192887086454240, −1.02082646472062891868015763374, 1.02082646472062891868015763374, 1.70764937931677192887086454240, 2.67917047695031923373062108827, 3.60757675599384505485592016635, 4.00806065561990963619132795847, 5.10924164426307824879037252386, 5.92686981145048990162068997783, 6.30696350075628989789053458923, 7.33281687794757748950948333520, 7.64056190935897453511080817401

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