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 − 2.45·5-s + 3.23·7-s + 9-s + 0.0433·11-s + 3.92·13-s − 2.45·15-s − 6.01·17-s − 4.13·19-s + 3.23·21-s + 23-s + 1.00·25-s + 27-s − 29-s + 2.00·31-s + 0.0433·33-s − 7.93·35-s + 3.70·37-s + 3.92·39-s + 9.70·41-s + 9.70·43-s − 2.45·45-s − 2.20·47-s + 3.47·49-s − 6.01·51-s − 8.20·53-s − 0.106·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.09·5-s + 1.22·7-s + 0.333·9-s + 0.0130·11-s + 1.08·13-s − 0.632·15-s − 1.45·17-s − 0.948·19-s + 0.706·21-s + 0.208·23-s + 0.201·25-s + 0.192·27-s − 0.185·29-s + 0.359·31-s + 0.00753·33-s − 1.34·35-s + 0.608·37-s + 0.627·39-s + 1.51·41-s + 1.48·43-s − 0.365·45-s − 0.321·47-s + 0.496·49-s − 0.842·51-s − 1.12·53-s − 0.0143·55-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  =  $(2,\ 8004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.345775669$
$L(\frac12)$  $\approx$  $2.345775669$
$L(\frac{3}{2})$   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 \( 1 + 2.45T + 5T^{2} \)
7 \( 1 - 3.23T + 7T^{2} \)
11 \( 1 - 0.0433T + 11T^{2} \)
13 \( 1 - 3.92T + 13T^{2} \)
17 \( 1 + 6.01T + 17T^{2} \)
19 \( 1 + 4.13T + 19T^{2} \)
31 \( 1 - 2.00T + 31T^{2} \)
37 \( 1 - 3.70T + 37T^{2} \)
41 \( 1 - 9.70T + 41T^{2} \)
43 \( 1 - 9.70T + 43T^{2} \)
47 \( 1 + 2.20T + 47T^{2} \)
53 \( 1 + 8.20T + 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 - 4.07T + 61T^{2} \)
67 \( 1 - 0.00749T + 67T^{2} \)
71 \( 1 - 9.37T + 71T^{2} \)
73 \( 1 + 3.48T + 73T^{2} \)
79 \( 1 - 4.80T + 79T^{2} \)
83 \( 1 + 14.3T + 83T^{2} \)
89 \( 1 - 2.96T + 89T^{2} \)
97 \( 1 - 3.26T + 97T^{2} \)
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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

−7.979489921867458900733407173206, −7.34213912332429180273096997736, −6.56834012868323747969131761716, −5.80089952336529778680167412134, −4.67168193678533144202161414089, −4.26891430543356663441704551545, −3.72468726239152283164021079261, −2.60852319434506644257113631293, −1.85434725800220879732006596321, −0.74735741452627708571588207390, 0.74735741452627708571588207390, 1.85434725800220879732006596321, 2.60852319434506644257113631293, 3.72468726239152283164021079261, 4.26891430543356663441704551545, 4.67168193678533144202161414089, 5.80089952336529778680167412134, 6.56834012868323747969131761716, 7.34213912332429180273096997736, 7.979489921867458900733407173206

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