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 − 2.34·5-s − 1.25·7-s + 9-s − 6.44·11-s + 1.34·13-s − 2.34·15-s − 7.39·17-s − 4.27·19-s − 1.25·21-s + 23-s + 0.499·25-s + 27-s − 29-s + 10.1·31-s − 6.44·33-s + 2.95·35-s − 5.20·37-s + 1.34·39-s − 10.6·41-s + 0.118·43-s − 2.34·45-s + 9.08·47-s − 5.41·49-s − 7.39·51-s + 10.4·53-s + 15.1·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.04·5-s − 0.475·7-s + 0.333·9-s − 1.94·11-s + 0.373·13-s − 0.605·15-s − 1.79·17-s − 0.981·19-s − 0.274·21-s + 0.208·23-s + 0.0998·25-s + 0.192·27-s − 0.185·29-s + 1.81·31-s − 1.12·33-s + 0.498·35-s − 0.856·37-s + 0.215·39-s − 1.65·41-s + 0.0180·43-s − 0.349·45-s + 1.32·47-s − 0.773·49-s − 1.03·51-s + 1.43·53-s + 2.03·55-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  =  $(2,\ 8004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7081469475$
$L(\frac12)$  $\approx$  $0.7081469475$
$L(\frac{3}{2})$   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 \( 1 + 2.34T + 5T^{2} \)
7 \( 1 + 1.25T + 7T^{2} \)
11 \( 1 + 6.44T + 11T^{2} \)
13 \( 1 - 1.34T + 13T^{2} \)
17 \( 1 + 7.39T + 17T^{2} \)
19 \( 1 + 4.27T + 19T^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
37 \( 1 + 5.20T + 37T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 - 0.118T + 43T^{2} \)
47 \( 1 - 9.08T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + 3.39T + 59T^{2} \)
61 \( 1 + 2.23T + 61T^{2} \)
67 \( 1 - 7.85T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 - 7.17T + 73T^{2} \)
79 \( 1 + 4.24T + 79T^{2} \)
83 \( 1 + 7.91T + 83T^{2} \)
89 \( 1 + 4.52T + 89T^{2} \)
97 \( 1 + 0.279T + 97T^{2} \)
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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

−7.984890516468071692518799725578, −7.16962430224819655163308201648, −6.68355369233497180668056127757, −5.77171701971693181523364907968, −4.77527350290997711756463512524, −4.31730943553791160291080596807, −3.44659722816928545714425508081, −2.71242766560783337534492626997, −2.04797189432331332621075817279, −0.37519141919393307256627573764, 0.37519141919393307256627573764, 2.04797189432331332621075817279, 2.71242766560783337534492626997, 3.44659722816928545714425508081, 4.31730943553791160291080596807, 4.77527350290997711756463512524, 5.77171701971693181523364907968, 6.68355369233497180668056127757, 7.16962430224819655163308201648, 7.984890516468071692518799725578

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