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 + 1.72·5-s + 4.78·7-s + 9-s − 0.855·11-s + 0.789·13-s + 1.72·15-s − 0.187·17-s + 2.96·19-s + 4.78·21-s + 23-s − 2.02·25-s + 27-s − 29-s + 6.56·31-s − 0.855·33-s + 8.24·35-s + 6.29·37-s + 0.789·39-s − 5.32·41-s + 4.21·43-s + 1.72·45-s − 5.43·47-s + 15.8·49-s − 0.187·51-s + 0.912·53-s − 1.47·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.771·5-s + 1.80·7-s + 0.333·9-s − 0.257·11-s + 0.218·13-s + 0.445·15-s − 0.0453·17-s + 0.680·19-s + 1.04·21-s + 0.208·23-s − 0.405·25-s + 0.192·27-s − 0.185·29-s + 1.17·31-s − 0.148·33-s + 1.39·35-s + 1.03·37-s + 0.126·39-s − 0.831·41-s + 0.643·43-s + 0.257·45-s − 0.793·47-s + 2.27·49-s − 0.0261·51-s + 0.125·53-s − 0.198·55-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  =  $(2,\ 8004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.189889282$
$L(\frac12)$  $\approx$  $4.189889282$
$L(\frac{3}{2})$   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 \( 1 - 1.72T + 5T^{2} \)
7 \( 1 - 4.78T + 7T^{2} \)
11 \( 1 + 0.855T + 11T^{2} \)
13 \( 1 - 0.789T + 13T^{2} \)
17 \( 1 + 0.187T + 17T^{2} \)
19 \( 1 - 2.96T + 19T^{2} \)
31 \( 1 - 6.56T + 31T^{2} \)
37 \( 1 - 6.29T + 37T^{2} \)
41 \( 1 + 5.32T + 41T^{2} \)
43 \( 1 - 4.21T + 43T^{2} \)
47 \( 1 + 5.43T + 47T^{2} \)
53 \( 1 - 0.912T + 53T^{2} \)
59 \( 1 + 4.84T + 59T^{2} \)
61 \( 1 + 7.62T + 61T^{2} \)
67 \( 1 + 12.2T + 67T^{2} \)
71 \( 1 + 5.33T + 71T^{2} \)
73 \( 1 + 1.62T + 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 - 8.62T + 89T^{2} \)
97 \( 1 + 6.23T + 97T^{2} \)
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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

−7.79440682351328643651577218396, −7.47876089371025374860385556503, −6.36781907512513285291735054849, −5.72582303068312331363322330251, −4.88141548429127254417999993439, −4.51850494591490759988192797091, −3.43219571022189260145436343095, −2.49864224231134284454473828892, −1.80459474103182268544735751075, −1.08445093557216603842412879363, 1.08445093557216603842412879363, 1.80459474103182268544735751075, 2.49864224231134284454473828892, 3.43219571022189260145436343095, 4.51850494591490759988192797091, 4.88141548429127254417999993439, 5.72582303068312331363322330251, 6.36781907512513285291735054849, 7.47876089371025374860385556503, 7.79440682351328643651577218396

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