Properties

Degree 2
Conductor $ 2 \cdot 4001 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2.76·3-s + 4-s + 4.07·5-s − 2.76·6-s + 2.61·7-s + 8-s + 4.66·9-s + 4.07·10-s − 4.42·11-s − 2.76·12-s − 6.42·13-s + 2.61·14-s − 11.2·15-s + 16-s − 4.31·17-s + 4.66·18-s − 3.26·19-s + 4.07·20-s − 7.24·21-s − 4.42·22-s + 3.62·23-s − 2.76·24-s + 11.6·25-s − 6.42·26-s − 4.60·27-s + 2.61·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.59·3-s + 0.5·4-s + 1.82·5-s − 1.13·6-s + 0.989·7-s + 0.353·8-s + 1.55·9-s + 1.28·10-s − 1.33·11-s − 0.799·12-s − 1.78·13-s + 0.699·14-s − 2.91·15-s + 0.250·16-s − 1.04·17-s + 1.09·18-s − 0.748·19-s + 0.911·20-s − 1.58·21-s − 0.943·22-s + 0.755·23-s − 0.565·24-s + 2.32·25-s − 1.25·26-s − 0.885·27-s + 0.494·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8002\)    =    \(2 \cdot 4001\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8002} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 8002,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;4001\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;4001\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
4001 \( 1+O(T) \)
good3 \( 1 + 2.76T + 3T^{2} \)
5 \( 1 - 4.07T + 5T^{2} \)
7 \( 1 - 2.61T + 7T^{2} \)
11 \( 1 + 4.42T + 11T^{2} \)
13 \( 1 + 6.42T + 13T^{2} \)
17 \( 1 + 4.31T + 17T^{2} \)
19 \( 1 + 3.26T + 19T^{2} \)
23 \( 1 - 3.62T + 23T^{2} \)
29 \( 1 + 1.10T + 29T^{2} \)
31 \( 1 - 10.0T + 31T^{2} \)
37 \( 1 + 1.48T + 37T^{2} \)
41 \( 1 + 2.84T + 41T^{2} \)
43 \( 1 + 9.08T + 43T^{2} \)
47 \( 1 + 5.38T + 47T^{2} \)
53 \( 1 + 8.55T + 53T^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 + 6.07T + 61T^{2} \)
67 \( 1 + 7.23T + 67T^{2} \)
71 \( 1 - 3.42T + 71T^{2} \)
73 \( 1 + 16.3T + 73T^{2} \)
79 \( 1 - 0.250T + 79T^{2} \)
83 \( 1 - 12.3T + 83T^{2} \)
89 \( 1 + 6.26T + 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−7.08367349678861348407097613455, −6.52368360607952650422521806819, −6.01347236680435850650389022222, −5.13526570395820117501267076497, −4.91799999649628774577825459762, −4.67322932284053069921435014903, −2.78918721800251249635214510520, −2.22340670172562746361305451741, −1.44575417452835789177766471701, 0, 1.44575417452835789177766471701, 2.22340670172562746361305451741, 2.78918721800251249635214510520, 4.67322932284053069921435014903, 4.91799999649628774577825459762, 5.13526570395820117501267076497, 6.01347236680435850650389022222, 6.52368360607952650422521806819, 7.08367349678861348407097613455

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