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 − 1.86·5-s − 4.84·7-s + 9-s − 4.93·11-s + 2.86·13-s − 1.86·15-s + 3.10·17-s − 1.94·19-s − 4.84·21-s + 23-s − 1.51·25-s + 27-s − 29-s − 5.41·31-s − 4.93·33-s + 9.04·35-s − 6.81·37-s + 2.86·39-s + 10.1·41-s − 3.21·43-s − 1.86·45-s − 12.3·47-s + 16.4·49-s + 3.10·51-s − 13.4·53-s + 9.21·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.835·5-s − 1.83·7-s + 0.333·9-s − 1.48·11-s + 0.793·13-s − 0.482·15-s + 0.753·17-s − 0.447·19-s − 1.05·21-s + 0.208·23-s − 0.302·25-s + 0.192·27-s − 0.185·29-s − 0.973·31-s − 0.859·33-s + 1.52·35-s − 1.12·37-s + 0.458·39-s + 1.59·41-s − 0.490·43-s − 0.278·45-s − 1.79·47-s + 2.35·49-s + 0.435·51-s − 1.84·53-s + 1.24·55-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  =  $(2,\ 8004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7438362505$
$L(\frac12)$  $\approx$  $0.7438362505$
$L(\frac{3}{2})$   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 \( 1 + 1.86T + 5T^{2} \)
7 \( 1 + 4.84T + 7T^{2} \)
11 \( 1 + 4.93T + 11T^{2} \)
13 \( 1 - 2.86T + 13T^{2} \)
17 \( 1 - 3.10T + 17T^{2} \)
19 \( 1 + 1.94T + 19T^{2} \)
31 \( 1 + 5.41T + 31T^{2} \)
37 \( 1 + 6.81T + 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + 3.21T + 43T^{2} \)
47 \( 1 + 12.3T + 47T^{2} \)
53 \( 1 + 13.4T + 53T^{2} \)
59 \( 1 - 1.37T + 59T^{2} \)
61 \( 1 + 5.44T + 61T^{2} \)
67 \( 1 + 4.14T + 67T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 + 4.90T + 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 + 1.89T + 89T^{2} \)
97 \( 1 - 6.06T + 97T^{2} \)
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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

−7.74093281535400581730640030403, −7.36916111720708077418552429520, −6.44132893007584880153220621297, −5.90567194734341593221521772156, −5.00189584824769072787305900432, −4.03036013209725375481088776877, −3.21118321798125359126162283132, −3.14885013262761073997946394317, −1.90776562321353232815792303675, −0.39071070557498265506708825896, 0.39071070557498265506708825896, 1.90776562321353232815792303675, 3.14885013262761073997946394317, 3.21118321798125359126162283132, 4.03036013209725375481088776877, 5.00189584824769072787305900432, 5.90567194734341593221521772156, 6.44132893007584880153220621297, 7.36916111720708077418552429520, 7.74093281535400581730640030403

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