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 − 0.270·5-s + 0.409·7-s + 9-s − 3.18·11-s + 0.382·13-s − 0.270·15-s − 4.23·17-s + 5.44·19-s + 0.409·21-s + 23-s − 4.92·25-s + 27-s + 29-s + 2.08·31-s − 3.18·33-s − 0.110·35-s − 5.55·37-s + 0.382·39-s + 7.07·41-s − 5.46·43-s − 0.270·45-s − 11.6·47-s − 6.83·49-s − 4.23·51-s + 13.1·53-s + 0.861·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.121·5-s + 0.154·7-s + 0.333·9-s − 0.959·11-s + 0.106·13-s − 0.0699·15-s − 1.02·17-s + 1.24·19-s + 0.0892·21-s + 0.208·23-s − 0.985·25-s + 0.192·27-s + 0.185·29-s + 0.373·31-s − 0.553·33-s − 0.0187·35-s − 0.913·37-s + 0.0612·39-s + 1.10·41-s − 0.833·43-s − 0.0403·45-s − 1.70·47-s − 0.976·49-s − 0.592·51-s + 1.80·53-s + 0.116·55-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 \( 1 + 0.270T + 5T^{2} \)
7 \( 1 - 0.409T + 7T^{2} \)
11 \( 1 + 3.18T + 11T^{2} \)
13 \( 1 - 0.382T + 13T^{2} \)
17 \( 1 + 4.23T + 17T^{2} \)
19 \( 1 - 5.44T + 19T^{2} \)
31 \( 1 - 2.08T + 31T^{2} \)
37 \( 1 + 5.55T + 37T^{2} \)
41 \( 1 - 7.07T + 41T^{2} \)
43 \( 1 + 5.46T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 - 4.76T + 59T^{2} \)
61 \( 1 + 2.32T + 61T^{2} \)
67 \( 1 - 0.562T + 67T^{2} \)
71 \( 1 + 7.04T + 71T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 + 16.6T + 79T^{2} \)
83 \( 1 - 9.44T + 83T^{2} \)
89 \( 1 - 8.74T + 89T^{2} \)
97 \( 1 + 2.87T + 97T^{2} \)
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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

−7.54837484346023481190729641668, −6.98170240556482792206651671750, −6.12105604139399049691432521071, −5.28686806129007643823938778017, −4.69192721276418008809960885739, −3.82518081848001872219409461030, −3.04788441250730363541327967364, −2.33414834708812021740703344094, −1.38608665368054335285387118348, 0, 1.38608665368054335285387118348, 2.33414834708812021740703344094, 3.04788441250730363541327967364, 3.82518081848001872219409461030, 4.69192721276418008809960885739, 5.28686806129007643823938778017, 6.12105604139399049691432521071, 6.98170240556482792206651671750, 7.54837484346023481190729641668

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