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 − 1.84·5-s − 2.79·7-s + 9-s + 1.31·11-s + 2.49·13-s − 1.84·15-s − 3.40·17-s + 2.66·19-s − 2.79·21-s + 23-s − 1.60·25-s + 27-s + 29-s − 7.42·31-s + 1.31·33-s + 5.13·35-s + 2.47·37-s + 2.49·39-s + 5.01·41-s + 2.20·43-s − 1.84·45-s + 10.0·47-s + 0.789·49-s − 3.40·51-s − 0.225·53-s − 2.41·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.823·5-s − 1.05·7-s + 0.333·9-s + 0.395·11-s + 0.692·13-s − 0.475·15-s − 0.825·17-s + 0.612·19-s − 0.609·21-s + 0.208·23-s − 0.321·25-s + 0.192·27-s + 0.185·29-s − 1.33·31-s + 0.228·33-s + 0.868·35-s + 0.406·37-s + 0.399·39-s + 0.782·41-s + 0.335·43-s − 0.274·45-s + 1.47·47-s + 0.112·49-s − 0.476·51-s − 0.0309·53-s − 0.325·55-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 \( 1 + 1.84T + 5T^{2} \)
7 \( 1 + 2.79T + 7T^{2} \)
11 \( 1 - 1.31T + 11T^{2} \)
13 \( 1 - 2.49T + 13T^{2} \)
17 \( 1 + 3.40T + 17T^{2} \)
19 \( 1 - 2.66T + 19T^{2} \)
31 \( 1 + 7.42T + 31T^{2} \)
37 \( 1 - 2.47T + 37T^{2} \)
41 \( 1 - 5.01T + 41T^{2} \)
43 \( 1 - 2.20T + 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 + 0.225T + 53T^{2} \)
59 \( 1 + 5.31T + 59T^{2} \)
61 \( 1 - 11.3T + 61T^{2} \)
67 \( 1 - 4.47T + 67T^{2} \)
71 \( 1 + 3.68T + 71T^{2} \)
73 \( 1 + 9.11T + 73T^{2} \)
79 \( 1 + 1.28T + 79T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 + 0.382T + 97T^{2} \)
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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

−7.39310520589388870404730670226, −6.99573616733824058535401267768, −6.17594363359484094685207830444, −5.50487267139843380674341667125, −4.28760123666462986978779131053, −3.90198079681038360575980452291, −3.19886378678959776244596207190, −2.42014834836775487118337779404, −1.21180081081801366025377704891, 0, 1.21180081081801366025377704891, 2.42014834836775487118337779404, 3.19886378678959776244596207190, 3.90198079681038360575980452291, 4.28760123666462986978779131053, 5.50487267139843380674341667125, 6.17594363359484094685207830444, 6.99573616733824058535401267768, 7.39310520589388870404730670226

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