Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 401 $
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 − 1.88·3-s + 4-s − 5-s + 1.88·6-s + 1.11·7-s − 8-s + 0.563·9-s + 10-s − 6.32·11-s − 1.88·12-s − 2.82·13-s − 1.11·14-s + 1.88·15-s + 16-s + 3.93·17-s − 0.563·18-s − 1.08·19-s − 20-s − 2.11·21-s + 6.32·22-s + 5.37·23-s + 1.88·24-s + 25-s + 2.82·26-s + 4.59·27-s + 1.11·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.08·3-s + 0.5·4-s − 0.447·5-s + 0.770·6-s + 0.422·7-s − 0.353·8-s + 0.187·9-s + 0.316·10-s − 1.90·11-s − 0.544·12-s − 0.783·13-s − 0.298·14-s + 0.487·15-s + 0.250·16-s + 0.953·17-s − 0.132·18-s − 0.248·19-s − 0.223·20-s − 0.460·21-s + 1.34·22-s + 1.12·23-s + 0.385·24-s + 0.200·25-s + 0.554·26-s + 0.885·27-s + 0.211·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4010\)    =    \(2 \cdot 5 \cdot 401\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4010} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4010,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;401\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;401\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
5 \( 1 + T \)
401 \( 1 + T \)
good3 \( 1 + 1.88T + 3T^{2} \)
7 \( 1 - 1.11T + 7T^{2} \)
11 \( 1 + 6.32T + 11T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 - 3.93T + 17T^{2} \)
19 \( 1 + 1.08T + 19T^{2} \)
23 \( 1 - 5.37T + 23T^{2} \)
29 \( 1 - 3.07T + 29T^{2} \)
31 \( 1 - 3.88T + 31T^{2} \)
37 \( 1 + 0.283T + 37T^{2} \)
41 \( 1 - 11.9T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 + 6.23T + 47T^{2} \)
53 \( 1 + 0.293T + 53T^{2} \)
59 \( 1 - 6.60T + 59T^{2} \)
61 \( 1 - 11.7T + 61T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 - 4.56T + 71T^{2} \)
73 \( 1 + 16.1T + 73T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 + 6.28T + 83T^{2} \)
89 \( 1 - 7.99T + 89T^{2} \)
97 \( 1 + 8.56T + 97T^{2} \)
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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

−8.158256034550464702356660016279, −7.39026212021026847582299263128, −6.77542373434459617315707499581, −5.75995826338345456882641106336, −5.16073323250555564945427384268, −4.65359010862600743843223374229, −3.14438477323698152828718673937, −2.44228088787066898959849467176, −0.971848507756291376257836399583, 0, 0.971848507756291376257836399583, 2.44228088787066898959849467176, 3.14438477323698152828718673937, 4.65359010862600743843223374229, 5.16073323250555564945427384268, 5.75995826338345456882641106336, 6.77542373434459617315707499581, 7.39026212021026847582299263128, 8.158256034550464702356660016279

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