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 + 1.71·5-s + 0.210·7-s + 9-s − 2.37·11-s − 3.22·13-s + 1.71·15-s − 1.22·17-s − 7.29·19-s + 0.210·21-s + 23-s − 2.04·25-s + 27-s + 29-s + 3.68·31-s − 2.37·33-s + 0.361·35-s + 7.65·37-s − 3.22·39-s + 2.82·41-s + 9.02·43-s + 1.71·45-s − 5.43·47-s − 6.95·49-s − 1.22·51-s − 2.49·53-s − 4.07·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.768·5-s + 0.0793·7-s + 0.333·9-s − 0.715·11-s − 0.894·13-s + 0.443·15-s − 0.298·17-s − 1.67·19-s + 0.0458·21-s + 0.208·23-s − 0.408·25-s + 0.192·27-s + 0.185·29-s + 0.662·31-s − 0.413·33-s + 0.0610·35-s + 1.25·37-s − 0.516·39-s + 0.441·41-s + 1.37·43-s + 0.256·45-s − 0.792·47-s − 0.993·49-s − 0.172·51-s − 0.342·53-s − 0.550·55-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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 - T \)
good5 \( 1 - 1.71T + 5T^{2} \)
7 \( 1 - 0.210T + 7T^{2} \)
11 \( 1 + 2.37T + 11T^{2} \)
13 \( 1 + 3.22T + 13T^{2} \)
17 \( 1 + 1.22T + 17T^{2} \)
19 \( 1 + 7.29T + 19T^{2} \)
31 \( 1 - 3.68T + 31T^{2} \)
37 \( 1 - 7.65T + 37T^{2} \)
41 \( 1 - 2.82T + 41T^{2} \)
43 \( 1 - 9.02T + 43T^{2} \)
47 \( 1 + 5.43T + 47T^{2} \)
53 \( 1 + 2.49T + 53T^{2} \)
59 \( 1 + 3.55T + 59T^{2} \)
61 \( 1 + 7.27T + 61T^{2} \)
67 \( 1 - 9.40T + 67T^{2} \)
71 \( 1 + 7.53T + 71T^{2} \)
73 \( 1 - 3.75T + 73T^{2} \)
79 \( 1 - 0.807T + 79T^{2} \)
83 \( 1 - 5.58T + 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−7.71780819858286144563731701799, −6.69615536238952654517844446784, −6.22526858609929570752577706569, −5.36773294859359879627230164326, −4.60927789440921469308837766118, −4.01923417417396987408866195168, −2.68732456205018528481332524017, −2.47800045106545242599413016678, −1.48929284275036063428538481384, 0, 1.48929284275036063428538481384, 2.47800045106545242599413016678, 2.68732456205018528481332524017, 4.01923417417396987408866195168, 4.60927789440921469308837766118, 5.36773294859359879627230164326, 6.22526858609929570752577706569, 6.69615536238952654517844446784, 7.71780819858286144563731701799

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