Properties

Degree 2
Conductor $ 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  + 2-s + 3-s + 4-s + 4·5-s + 6-s − 0.449·7-s + 8-s + 9-s + 4·10-s − 4.89·11-s + 12-s + 2.89·13-s − 0.449·14-s + 4·15-s + 16-s + 4.89·17-s + 18-s + 0.449·19-s + 4·20-s − 0.449·21-s − 4.89·22-s − 23-s + 24-s + 11·25-s + 2.89·26-s + 27-s − 0.449·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.78·5-s + 0.408·6-s − 0.169·7-s + 0.353·8-s + 0.333·9-s + 1.26·10-s − 1.47·11-s + 0.288·12-s + 0.804·13-s − 0.120·14-s + 1.03·15-s + 0.250·16-s + 1.18·17-s + 0.235·18-s + 0.103·19-s + 0.894·20-s − 0.0980·21-s − 1.04·22-s − 0.208·23-s + 0.204·24-s + 2.20·25-s + 0.568·26-s + 0.192·27-s − 0.0849·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4002\)    =    \(2 \cdot 3 \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4002} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 4002,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(5.294503506\)
\(L(\frac12)\)  \(\approx\)  \(5.294503506\)
\(L(\frac{3}{2})\)   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 - T \)
good5 \( 1 - 4T + 5T^{2} \)
7 \( 1 + 0.449T + 7T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 - 2.89T + 13T^{2} \)
17 \( 1 - 4.89T + 17T^{2} \)
19 \( 1 - 0.449T + 19T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 0.898T + 41T^{2} \)
43 \( 1 - 7.55T + 43T^{2} \)
47 \( 1 - 4T + 47T^{2} \)
53 \( 1 - 0.898T + 53T^{2} \)
59 \( 1 - 4.89T + 59T^{2} \)
61 \( 1 + 6.89T + 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + 1.10T + 71T^{2} \)
73 \( 1 - 2.89T + 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 6.44T + 83T^{2} \)
89 \( 1 - 3.10T + 89T^{2} \)
97 \( 1 - 15.1T + 97T^{2} \)
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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

−8.403090421328483914470569760146, −7.68474069536071076522704141009, −6.86220626190112201930814571270, −5.93677371692860366359229954731, −5.57786271831326875210536455653, −4.89548307172694561832298614343, −3.67721095474791598088477050452, −2.86364388705708469923587278650, −2.21983711838196141782436989069, −1.29017147924949527991854214699, 1.29017147924949527991854214699, 2.21983711838196141782436989069, 2.86364388705708469923587278650, 3.67721095474791598088477050452, 4.89548307172694561832298614343, 5.57786271831326875210536455653, 5.93677371692860366359229954731, 6.86220626190112201930814571270, 7.68474069536071076522704141009, 8.403090421328483914470569760146

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