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 + 0.138·5-s − 3.33·7-s + 9-s − 2.06·11-s − 0.702·13-s + 0.138·15-s + 4.32·17-s + 2.44·19-s − 3.33·21-s + 23-s − 4.98·25-s + 27-s + 29-s + 7.41·31-s − 2.06·33-s − 0.462·35-s − 2.90·37-s − 0.702·39-s − 4.05·41-s + 0.235·43-s + 0.138·45-s − 2.05·47-s + 4.13·49-s + 4.32·51-s − 3.22·53-s − 0.286·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.0619·5-s − 1.26·7-s + 0.333·9-s − 0.622·11-s − 0.194·13-s + 0.0357·15-s + 1.04·17-s + 0.560·19-s − 0.728·21-s + 0.208·23-s − 0.996·25-s + 0.192·27-s + 0.185·29-s + 1.33·31-s − 0.359·33-s − 0.0781·35-s − 0.477·37-s − 0.112·39-s − 0.633·41-s + 0.0358·43-s + 0.0206·45-s − 0.299·47-s + 0.590·49-s + 0.605·51-s − 0.442·53-s − 0.0386·55-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 \( 1 - 0.138T + 5T^{2} \)
7 \( 1 + 3.33T + 7T^{2} \)
11 \( 1 + 2.06T + 11T^{2} \)
13 \( 1 + 0.702T + 13T^{2} \)
17 \( 1 - 4.32T + 17T^{2} \)
19 \( 1 - 2.44T + 19T^{2} \)
31 \( 1 - 7.41T + 31T^{2} \)
37 \( 1 + 2.90T + 37T^{2} \)
41 \( 1 + 4.05T + 41T^{2} \)
43 \( 1 - 0.235T + 43T^{2} \)
47 \( 1 + 2.05T + 47T^{2} \)
53 \( 1 + 3.22T + 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 + 2.78T + 61T^{2} \)
67 \( 1 + 11.6T + 67T^{2} \)
71 \( 1 - 7.47T + 71T^{2} \)
73 \( 1 - 9.16T + 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 - 3.05T + 89T^{2} \)
97 \( 1 - 2.74T + 97T^{2} \)
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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

−7.69676506297676817330748687948, −6.77175069762353648354717574600, −6.20758566833848920578150854031, −5.42069719547119284129037435638, −4.66458323920941180598129425459, −3.61957744589759668635710845991, −3.14503774895373652153171867080, −2.45492843636136302291258141979, −1.28444693408459198439036465212, 0, 1.28444693408459198439036465212, 2.45492843636136302291258141979, 3.14503774895373652153171867080, 3.61957744589759668635710845991, 4.66458323920941180598129425459, 5.42069719547119284129037435638, 6.20758566833848920578150854031, 6.77175069762353648354717574600, 7.69676506297676817330748687948

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