Properties

Degree 2
Conductor $ 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  + 2-s + 3-s + 4-s + 3·5-s + 6-s + 3·7-s + 8-s + 9-s + 3·10-s + 0.732·11-s + 12-s + 4.73·13-s + 3·14-s + 3·15-s + 16-s + 0.464·17-s + 18-s − 4.46·19-s + 3·20-s + 3·21-s + 0.732·22-s + 23-s + 24-s + 4·25-s + 4.73·26-s + 27-s + 3·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.34·5-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 0.333·9-s + 0.948·10-s + 0.220·11-s + 0.288·12-s + 1.31·13-s + 0.801·14-s + 0.774·15-s + 0.250·16-s + 0.112·17-s + 0.235·18-s − 1.02·19-s + 0.670·20-s + 0.654·21-s + 0.156·22-s + 0.208·23-s + 0.204·24-s + 0.800·25-s + 0.928·26-s + 0.192·27-s + 0.566·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4002\)    =    \(2 \cdot 3 \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4002} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4002,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.893697174$
$L(\frac12)$  $\approx$  $5.893697174$
$L(\frac{3}{2})$   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 \( 1 - T \)
3 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 + T \)
good5 \( 1 - 3T + 5T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 - 0.732T + 11T^{2} \)
13 \( 1 - 4.73T + 13T^{2} \)
17 \( 1 - 0.464T + 17T^{2} \)
19 \( 1 + 4.46T + 19T^{2} \)
31 \( 1 + 2.73T + 31T^{2} \)
37 \( 1 - 3.19T + 37T^{2} \)
41 \( 1 + 12.1T + 41T^{2} \)
43 \( 1 + 11.9T + 43T^{2} \)
47 \( 1 + 9.39T + 47T^{2} \)
53 \( 1 - 2.53T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 + 14.3T + 61T^{2} \)
67 \( 1 + 2.53T + 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 - 0.928T + 73T^{2} \)
79 \( 1 - 7.12T + 79T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 - 1.46T + 89T^{2} \)
97 \( 1 - 4T + 97T^{2} \)
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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

−8.521617222218064874829368939853, −7.79374147666536197674691367321, −6.70477771501253320983184293126, −6.24342437650037440482466507213, −5.38587185575190134227132702737, −4.75908611811333670646025170905, −3.84495065956566398078168293970, −2.98182890718561413265398533638, −1.75647714637458985883726461871, −1.62580224380410981962656552652, 1.62580224380410981962656552652, 1.75647714637458985883726461871, 2.98182890718561413265398533638, 3.84495065956566398078168293970, 4.75908611811333670646025170905, 5.38587185575190134227132702737, 6.24342437650037440482466507213, 6.70477771501253320983184293126, 7.79374147666536197674691367321, 8.521617222218064874829368939853

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